Jun 15
  1. Basic Linear Algebra And Matrix Analysis


  1. Linear algebra is a fantastic subject. On the one hand it is clean and beautiful. If you have three vectors in 12-dimensional space, you can almost see them. A combination like the first plus the second minus twice the third is harder to visualize, but may still be possible. I don't know if anyone can see all such combinations, but somehow (in this course) you begin to do it. Certainly the combinations of those three vectors will not fill the whole 12-dimensional space.....What  those combinations do fill out is something important and not just to pure mathematicians.That is the other side of linear algebra. It is needed and used. Ten years ago it was taught too abstractly, and the crucial importance of the subject was missed. Such a situation could not continue. Linear algebra has become as basic and as applicable as calculus, and fortunately it is easier.-from the preface of Linear Algebra and Its Applications, 3rd Edition by Gilbert Strang

  2. Matrix Algebra lecture notes and problems Matt Kerr University of Washington St.Louis Spring Semester 2011 (PG)(These are handwritten lecture notes and materials for a standard but comprehensive and well composed introductory linear algebra course. Matt Kerr is an exceptional teacher and mathematician and he has a host of lecture notes for courses he’s taught available online. I’ve included as many as I can find because I find them outstanding and demonstrative of great dedication to his students by the author. Like many teachers, he uses scans of students’ handwritten lecture notes. But what’s important here is that he chooses the best, most detailed notes and gives the students’ authorship credit at the site.  Sadly, many other scanned notes posted online are basically throwaway “chicken scratchings” so the professor can say he or she posted notes for students. It takes time and effort to select and scan notes for posting that are actually helpful for students and represent faithfully what was said in class. For this, Kerr gets all the credit in the world. As for the notes themselves, they are very well done and represent linear algebra from a variety of perspectives: geometric, algebraic and  analytic. They are careful and the theory of vector spaces and linear operators is present ,but the emphasis is on applications-which is a valid choice for a beginning course.(I prefer more of a theoretical bent where theory and applications are presented in equal measure, but that’s just me. )   As I said before, the content of the course is pretty standard: Linear systems, row-reduction, matrix equations, linear transformations, invertibility, rank and nullity, determinants, vector spaces, basis and dimension, eigenvectors and eigenvalues, diagonalization, inner products and norms, orthogonality, the Gram-Schmidt orthogonality algorithm, the spectral theorem and quadratic forms. But the delivery and development is superior, with many examples, historical and personal notes. One of the best things about these notes is how Kerr in class pointed out how the notes differ from the textbook in great detail when it’s relevant. I wish more professors took the time to do this instead of just tossing their favorite book at the students, assigning homework and taking off for their real job: research. (For a more theoretical presentation of linear algebra by Kerr, he has typeset and posted a book draft of the lectures for an honors linear algebra course he taught at UCLA several years ago, which we review here. ) Highly recommended.
  3. Matrix Computations Wen-Wei Lin Department of Mathematics National Tsing Hua University Hsinchu May 5, 2008(PG-13) These are the notes for a graduate course on numerical linear algebra, a subject that has grown in enormous importance in both computer science and applied mathematics in recent years. The notes assume a first course in linear algerbra (matrix computations, inner products, norms, etc) and a good background in theoretical calculus. Lin’s notes are all business- very curt, no historical background and very little visual insight. But they are clearly written with many nice proofs and examples. You could do a lot worse then learn the subject here. Matrix Algebra STEM program (G) This is a set of very elementary and well written set notes on matrix algebra for the STEM program. The bare bones of matrix manipulation is certainly a subject you’d like high school kids in this program (well, all high school kids, really, but these in particular)  before entering university and taking a serious linear algebra course or even a serious course on calculus of several variables-and these notes are about as perfect as you could imagine for such a course. Not only does it discuss- with many examples-the essentials of matrix algebra such as addition, multiplication and scalar multiplication, more importantly, it discusses the terminology that relates vectors to matrices which is so criticial to understanding the underlying theory and which usually is either taken for granted or brushed over far too quickly in linear algebra courses. Row and column vectors, linear independence, upper and lower triangular matrices, the transpose of a matrix, the determinant,diagonalization and eigenvalues and eigenvectors,  etc. are all discussed in great detail with many examples.  There are very few proofs, but that’s fine because that’s not the reason for these notes. They’re to supply the student with a good understanding of the mechanical aspects of matrix structure and manipulation that don’t usually need a deep understanding of the theory. And they do an outstanding job of this. I can’t imagine a better preparation for math majors about to take a first course in linear algebra at any level or a better primer of matrix algebra for other scientists. Highly recommended.
  4. Matrix Algebra for Business notes by Erin Pearse Cornell University  (G) A very applied plug-and-chug set of notes explaining the mechanics of matrices to business students and other non-scientists. I’m sure they’ll be thrilled that they don’t have to apply any critical thought in these notes. That way they can focus their time on cheating their way into Harvard Business School and then a Wall Street firm where they can begin stealing pensions…………….
  5. Notes on Matrix Theory by Christopher Beattie and John Rossi Virginia Tech 2012(PG)
  6. MATRIX THEORY Robert Wheeler Virginia Tech University  (PG-13) This is a pair of notes for linear algebra and matrix algebra courses at Virginia Tech and they are very interesting and productive to compare and contrast. The notes by Beattie and Rossi are heavily computational and applied, at least in the early chapters. They contain many of the standard theorems of linear algebra and basic matrix analysis, but there are virtually no proofs until vector spaces are defined in chapter 4. After this point, while the more difficult results are proved, most are either left as exercises or omitted altogether As a result, they really form more of a detailed outline for a full course in the subject in their current state.  In fairness to the authors, they write very well and with a great deal of care and rigor-examples are solidly integrated into the explainations of definitions. But there are just too many holes for students to read through them without pencil in hand. While this might form the basis of an exciting Moore method type honors course for very strong students, most instructors would have to fill in details, particularly in more sophisticated topics such as eigenvalues and Jordan decomposition. In many ways, that’s what Wheeler’s handwritten, very detailed notes try to do-and they do a very good job indeed. These notes- all 546 pages of them! – fill in many of the details missing in Beattie and Rossi’s text, but this is just the beginning of what they  do. In many ways, Wheeler uses their notes as a starting point to develop his own class text and takes the material in his own unique direction. This is clearly a course intending to gird mathematics majors for serious coursework in abstract mathematics, particularly abstract algebra courses. Wheeler develops the theory of matricies and linear maps axiomatically, similar to an abstract algebra course. For example, he develops invertible matrices in terms of left and right inverses. Indeed,he doesn’t even assume a left (right) inverse is unique! Instead he begins with this minimalist definition and then proceeds to prove there is a single 2 sided inverse matrix for any invertable matrix. I first saw this construction in an undergraduate group theory course and it was critical for the development of my logical skills-I think in the concrete setting of matrices, it would be even more beneficial. The notes have tons of examples and pictures and cover all the ground in loving detail. And in addition, his handwriting is exceptional and very personable-they still have that wonderful human quality that lecture notes lose when they become full blown books. Wheeler has not only written a terrific expansion of Beattie and Rossi’s already solid notes, his notes are a great online linear algebra textbook in it’s own right. I don’t know if he plans at some point to turn it into an actual book, but if he does, it’ll be near the top of my list of choices as a free classroom text. In the meantime, I’ll be using it wholeheartedly as a source for both myself and my students. Beattie/ Rossi’s notes are solid and recommended, but the real gem here is Wheeler’s. Very highly recommended.
  7. Vectors & Matrices Stephen Cowley University of Cambridge Mathematical Tripos: IA (PG) An intensive course in concrete linear algebra intended for students in applied mathematics and physics at Cambridge. Cowley brings the same humor and depth to these notes that he brings to his vector analysis notes discussed above, although there is considerably more mathematical rigor and precision here then in those notes.  However,  this is certainly not a purely mathematical treatment of vector spaces, as difficult or subtle proofs-such as the proofs that every basis of a vector space has the same number of elements -are skipped. Crowley refers to its companion course, Linear Algebra I-II at Oxford’s notes for the full proofs.  It’s fascinating to me how in European courses, mathematical rigor is valued even in applied courses. You’ll rarely see a course in a U.K. based mathematics department that is purely practical, algorithmic and completely omits proofs the way “methods” courses in America do-unless it’s a graduate pure mathematics course and the proofs of all results are deliberately omitted and left for the students to prove! For this reason alone, the notes online at Cambridge and Oxford are a treasure to serious mathematics students of all interests. Most of the raw material of linear algebra is developed in the context of low dimensional spaces, such as solving systems of linear equations, linear independence, rank and nullity of matrices, eigenvalues and eigenvectors. There are also more practical topics of interest more to the applied mathematician then the pure algebraicist, such as complex numbers, the real and complex exponential and their matrix forms, conic sections and quadratic forms. There is an excellent section on tensor summation notation that everyone regardless of interest should read-this critical and confusing topic is so rarely explained this well. The notes are a joy to read and work through, as Cowley’s entertaining style shines through in every paragraph and example. A terrific example is given in a footnote to the discussion of the Einstien summation convention:Learning to omit the explicit sum is a bit like learning to change gear when starting to drive. At first you have to remind yourself that the sum is there, in the same way that you have to think consciously where to move gear knob. With practice you will learn to note the existence of the sum unconsciously, in the same way that an experienced driver changes gear unconsciously; however you will crash a few gears on the way!Constantly entertaining and enormously informative, these notes coupled with the aforementioned “abstract” course notes, they will form the basis (ugh! Unintentional pun!) for as comprehensive and informative a linear algebra course as one can find anywhere. Highly recommended.
  8. Vector Spaces Math1553 Ambar Sengupta Louisana State University Fall 2009 (PG-13) These are the supplemental notes on linear algebra that Sengupta wrote for the second half of the 2 year long honors calculus course at LSU. As such, they are quite a bit more incomplete in coverage then one looking for a general source would want-they basically cover just what’s needed for a rigorous development of multivariable calculus.  That being said, they are very clear and  rigorous given their limited scope with lots of pictures. Also, they lucidly present some of the basic topics of linear algebra that are usually reserved for more advanced courses with examples in low dimensions, such as the wedge product and the Hodge operator in R.both of which are used to give a careful definition of the cross product. Recommended as a supplementary resource, but you’ll have to go elsewhere for a complete course
  9. .Linear Systems Richard Froese and Brian Wetton Richard Froese and Brian Wetton University of British Columbia Spring, 2013   (PG) This is a very solid set of notes introducing linear algebra through the transformations of classical Euclidean geometry in R2  and R .  Matrix  computations as operations on their row and column spaces are emphasized, giving a strongly geometric flair to even the most abstract concepts. Many applications to both geometry and physics are given as well and both algebraic and geometric aspects are discussed in a unified manner, with matrices defined over both the real and complex fields. For example, the dot product in Rand its relationship to the area of a parelleogram in the plane is used to motivate and construct the determinant, which is subsequently generalized to Rvia the scalar triple product and the parallelogram box construction. The use of complex scalars allows a more general approach that illustrates important concepts that are sometimes skipped in a first course, such as the eigenvalue solutions of the complex characteristic equation of the rotation matrices in plane and 3-space. A masterly first course in the subject that will prepare most mathematics majors for more sophisticated treatments and applications of this critical subject in both abstract algebra and differential geometry.  Highly recommended.  
  10. Linear Algebra I Francis Wright Queen Mary College University of London 2005(PG-13)  Universities in the United Kingdom are one of my favorite places for obtaining study materials because they seem to value university teaching of mathematics more then just about anyone else. These wonderful notes for a first course in linear algebra is an excellent example. They are concise and to the point, yet still have many examples and pictures. More importantly, they develop linear algebra in the context of introducing the basic concepts of abstract algebra, such as groups and fields, with special emphasis on the properties of a vector space that relate it to them-namely, that every vector space is an additive Abelian group over a field of scalars. All the standard topics are covered clearly, carefully and beautifully with most examples drawn from classical geometry of the plane and 3-space: vector spaces, subspaces and their operations, linear independence and bases, dimension, linear operators and basis/coordinate system transformations, the rank of linear maps and the determinant function, eigenvectors and eigenvalues, orthogonal spaces and inner products. The MAPLE computer algebra system is used throughout to construct examples and drawings, introducing a powerful tool. The notes constantly try and describe geometrically the abstract concepts, sometimes in surprisingly innovative ways. For example, eigenvectors are introduced as the images of rotations of unit vectors. The combination of careful rigor at an elementary level, many examples and pictures and the relative brevity of the notes makes this one of my favorite sources and I’d be thrilled teaching a bunch of sophomores with it. Highly recommended!
  11.  Linear Algebra Peter J. Cameron Queen Mary College University of London 2010 (PG-13) Ironically, these are the notes to the follow up course to the  one taught by Wright immediately preceding it. Cameron is probably known to many as an expert in combinatorics, but he’s taught all manner of courses at Queen Mary and he’s written many resulting lecture notes. Although this is a second course on the subject and it does assume some familiarity with matrices, linear maps and vector spaces, it does give a rapid review of the basics to make them self contained. Frankly, though, even with honors students, I’d be really leery to do that and I’d insist on them being familiar with the basics before taking a course based on them or using them for self study. The notes are more abstract then Wright’s-they cover all the material in those notes quickly and proceed to go into considerably deeper waters. There are much fewer pictures, although Cameron does give a relatively large number of examples for a course at this level. The emphasis here-as is appropriate for a second course-is on abstract rigor with applications kept to a minimum.Projections, linear and quadratic forms, adjoint operators, inner products and orthonormal bases, symmetric and skew-symmetric linear operators and Hermitian matrices in both the real and complex cases are fully discussed. The normal form is introduced much earlier, although the Jordan form is stated without proof. The main flaw in these otherwise very solid notes are there are no explicit exercises, although  many minor results are stated without proof. This is probably due to the U.K. educational structure, which usually posts exercises and exams separately from notes.  Making those available at his website would make these notes much easier to use. Still, exercises at this level shouldn’t be too difficult to find-Erdman’s collection would be quite helpful here. All in all, a very good second course in the subject and paired with the Wright notes above (or a comparable substitute), they can form the core of an extremely strong first year of linear algebra for serious mathematics students. Highly recommended.
  12. Elementary Linear Algebra Kuttler Bringham Young University April 13, 2013(PG) Elementary LinearAlgebra Problems K.Kuttler BYU 2013(PG)
  13. ElementaryLinearAlgebra Complete Solutions for Problems K.Kuttler BYU 2013(PG)
  14. Linear Algebra, Theory And Applications (complete version of text) Kenneth Kuttler BYU November 5, 2013    (PG-13)These are 4 texts written for various levels of linear algebra course by Kenneth Kuttler at BYU. Actually, the first three texts are parts of a single textbook, containing respectively, the textbook proper, its problems and complete solutions, so there’s really only 2 textbooks here. The philosophy behind the totality of these texts is basically the same as those for the various versions of his calculus notes available at his homepage: to give a more or less complete presention of the subject with both theory and applications. Kuttler wrote 2 versions of the text with this same philosophy in order to make each accessible for students of various levels of preparation. The “Elemenrary” version covers the subjects one would expect in a rigorous 2nd or 3rd year linear algebra course for mathematics majors and serious students of the physical and social sciences: Vector algebra focusing on the dot and vector product in R2 and R3 respectively and applications to classical physics, systems of linear equations, basic matrix algebra, transpose and invertablity, column and row matricies, determinants, rank and nullity of matrices, linear transformations, the LU matrix factorization, eigenvalues and eigenvectors, orthogonal matricies and transformations. The book concludes with 3 chapters on abs tract vector spaces, inner product spaces and the relationship between matrices and their linear transformations. Notice the abstract framework of linear transformations is delayed until the end of the book, using the enormous development of concrete vector spaces through matrices, linear maps and number systems in the preceding chapters as a foundation for the subsequent abstraction. Also, applications are quite important in the “Elementary” version of the textbook-there are entire chapters on linear programming, numerical methods of solutions of linear systems and characteristic equations. The more sophisticated version-which we’ll call LAWTAA-is designed for a more advanced audience, such as honors student undergraduates or first year graduate students looking for a comprehensive source to strengthen their background in this critical subject. The “larger” book isn’t dramatically different in content from the “elementary” version, but its organization and emphasis does make it quite a bit more sophisticated. It adds some 60 pages of material and the optional sections of the elementary version are required in the longer version. The main difference between the 2 is that the relationship between vector spaces and abstract algebraic structures is much more heavily emphasized in the longer version, as would be appropriate in a more theoretical treatment. This is not to say the short version is a sloppy “practical” course in contrast with the longer version- on the contrary, it is equally careful, with all definitions, major theorems and sophisticated examples proven carefully. For example, there’s a full chapter in “Elementary” on the rigorous theory of determinants as permutations of ordered lists of entries in matrices that assign either 0,1 or -1. It’s probably the single most complete and understandable presentation of the determinant function I’ve ever seen. But this chapter is optional in the Elementary version-where this entire chapter is required and presented early in the advanced version. Applications are downplayed in LAWTAA and topics not present in the “easier” version are focused on, such as complete fields, canonical forms, abstract inner product spaces, self-adjoint operators, norms, positive matrices and the theory of linear spaces of differential equations. As I’ve said, Kuttler’s goal with these books was similar to that of his calculus books, but I think he’s been even more successful
  15. with his treatments of linear algebra. Both books are extremely well organized, have a terrific choice of topics, are beautifully and clearly written with many unusal examples and are pitched perfectly for courses at their intended levels. In short, he’s written 2 outstanding books that all students and teachers of linear algebra should familiarize themselves with and download before they’re published and we lose free access to these jewels. I have no doubt one day, they’ll take their place on bookshelfs alongside Strang, Hoffman/Kunze and Curtis as classics of a subject that sadly, isn’t usually taken seriously enough by either students or teachers in its importance. Very highly recommended
  16. .Linear Algebra II Lecture Notes John C. Bowman University of Alberta 2010 (PG-13)  This is kind of a strange course to American eyes since it’s labeled “Linear Algebra II”-meanwhile, it begins at the very beginning with low dimensional vectors and moves on to the standard material of a linear algebra course in abstract vector spaces. My guess-from the online syllabus for the course taught this winter-is that these are actually the notes for both the first and second semesters of linear algebra and that the actual course begins with the complex numbers. In any event, these notes are extremely terse and omit virtually all proofs-they aren’t so much a set of lecture notes as an outline of what’s to be covered in this year-long course. The result isn't nearly as useful or readable as Bowman's honors and advanced calculus notes. Still, advanced students or graduate students getting ready for qualifying exams will find them useful if they review by attempting to fill in all the missing details. Lesser students will probably find them useful as a paperweight and little else.
  17. Linear Algebra Alexei Novikov Penn State University 2003(PG)   A concise set of notes to supplement a course based on Strang's classic text. Nice and readable, but not much meat or depth. They're not really intended to replace the textbook,after all. Ok but nothing awesome.
  18. Linear  algebra and differential equations Ryan Reich Harvard/University of Michigan University Spring 2010: (PG-13)  A very broad and rapid, but quite lucid presentation of the elements of linear algebra. Riech moves very quickly from Guass-Jordan elimination and solution of linear systems to linear transformations to vector spaces and their subspaces to orthogonality to determinants to spectral theory and applications to differential equations. The notes end with a brief discussion of infinite Fourier series. Each section is incredibly brief, but crystal clear with very good and modern examples.There’s also some unusual applications to stability theory, error correcting codes, least squares analysis and partial differential equations. A very nice set of notes, but the brevity and advanced level of them would make me seek out additional sources to use in conjunction with them. I think they’d also be very useful to graduate students looking to quickly review and/or cement their knowledge of linear algebra. Recommended.
  19.  LINEAR ALGEBRA CHRIS WOODWARD Rutgers University 2002 (PG-13)  This is an excellent unified presentation of linear algebra in the context of classical Euclidean geometry. Matricies, determinants, linear transformations, vector spaces and their subspaces, orthogonality and inner products, spectral theory (i.e. eigenvalues and eigenvectors), diagonalization theory,symmetric matrices and quadratic forms. The notes conclude with an interesting brief discussion of dynamical  systems. Many linear transformations and their matrices are explicated  in terms of the similarities, isometries and projection maps of Euclidean space. For example, the rotation matrix in R2 is discussed several times and in several different contexts, such as determinants and vector rotation in the plane. It is very geometric, with many excellent and informative pictures. It also contains a surprisingly large number of examples for such a relatively brief (62 pages) set of notes. The geometry of diagonalization and spectral theory are both also discussed rigorously and in greater depth then usual. Woodward also manages to squeeze in some interesting and unusual applications, such as a dynamical systems model of a disease epidemic in a fixed population.  I was really surprised how much I liked these notes and I almost bypassed them because I generally don’t like Postscript files-they’re tricky to open and read sometimes. That’s a shame since some of the est lectures on the web are in postscript-like these.An outstanding and inventive introduction to linear algebra. Very highly recommended.
  20. Linear Algebra by WWL Chen  (PG) I was really excited to find that Chen had written up his notes to a linear algebra course he had taught. I was really hopful they were every bit as sophisticated and wonderful as his other notes. I’m happy to report they indeed are.  Like his calculus notes, the presentation is a wonderful balance of theory and applications and the coverage is somewhat more then the average U.S. first course. Linear systems and matrices, row and reduced row echelon forms, elementary row operations, inverse and identity matrices, determinants, the geometry and algebra of vectors in Rand Rvector spaces, linear independence and bases, dimension, row and column spaces, rank and nullity, eigenvalues and eigenspaces, linear transformations and their relationship to matrices, real and complex Euclidean inner products spaces, orthogonal maps and spaces and quadratic forms. Not only are all major theorems proves clearly and concisely, not only are there many step by step computations and examples, but there are a host of applications given to the physical and social sciences-and a number of them are very nonstandard and uncommon in a first course. To give several examples, in addition to the usual applications of vectors and vector spaces to electrical networks, mechanics and network flows, Chen considers the use of elementary matrices to balance systems of chemical reactions, the use of the diagonalization product in genetics to solve for the the probabilities of  the autosomal phenotype of the nth generation of a plant breeding cycle, applications of matrix algebra to computer graphics and game theory. Yes, you read that second one correctly-and if you haven’t seen it before, you have to see it to believe it. Most incredibly-this marvelous course in total comes in at 244 pages when all pdf files are merged. You may-repeat, MAY-be able to find a more beautifully written and organized first course in linear spaces-but I dare anyone to find one that accomplishes all Chen does in these notes at that length. One of the very best sets of notes on any topic anywhere. I give them the highest possible recommendation for either teachers or students at any level.
  21. Linear Algebra Carl de Boor University of Wisconson-Madison draft 25 jan 2013 (PG-13) This is the draft of a book de Boor is working on based on the advanced linear algebra course he’s taught several times to advanced undergraduate math majors and graduate students in the physical sciences. I’ve found the notes for advanced linear algebra courses vary dramatically across universities depending on the insights of the professor and the makeup of the audience he or she is constructing the course for.de Boor states his goals as follows: This book is motivated by the following realizations:(1) The linear maps between a vector space X over the scalar field IF and  the associated coordinate spaces  IFn are efficient tools for work on theoretical and practical problems involving X. Those from IFn to X share with matrices the feature of columns, hence are called column maps, while those from Xto IFn share with matrices the feature of rows, hence are called row maps. Work with a linear map A usually requires its factorization into a column map and a row map. Such factorization is most efficient for the task if the particular column map is invertible as a map to the range of A, i.e., if it is a basis for that range.(2) Gauss elimination is applied to matrices for the purpose of obtaining bases for their nullspace and for their range. It results in a sequence of matrices all with the same nullspace, with the last matrix making the nullspace quite evident(3) A change of basis amounts to interpolation and vice versa. (4) Since the eigenstructure of a linear map A on a vector space X over  the scalar field IF is of interest in the study of the sequence A0 = id,A1 = A,A2, . . . of the powers of A, its derivation and discussion is best handled in terms of polynomials p(A) in that linear map with coefficients in IF. While determinants are indispensible and powerful tools in certain situations, they do not provide the best path to understanding eigenstructure.(5) In applications, vector spaces are, by and large, spaces of maps with the vector operations defined pointwise, or derived from such spaces in a straightforward manner. The coordinate spaces IFn are merely the simplest examples of such vector spaces.    De Boor follows this game plan through the entire manuscript, focusing on the function space approach to vector spaces and expressing many of the more sophisticated concepts of linear algebra via “approximation” of linear maps by sequences of simpler maps and /or matrices It’s very surprising how much insight can be gained when using this perspective for concepts that are useful in analysis and elementary geometry-although I’m not certain if this viewpoint would be equally as effective in more sophisticated concepts like differential forms. The notes are deep and dense, with many historical anecdotes and unusual approaches to the material. For example, his introduction of the rigorous definition of lists ( basically, finite indexed sets)  will be of great help to computer science majors and allows de Boor to use the implementation of matrices as ordered lists from the very beginning. It also allows for a rigorous definition of matrices, which even advanced treatments usually enshew-essentially, they are lists whose domains are Cartesian products of 2 finite subsets of the natural numbers and whose ranges lie in a field.  The standard topics of a linear algebra course are covered, albeit more rigorous and abstractly then usual, followed by more advanced topics such as quadratic forms  There are many examples and proofs and the “approximation” viewpoint is remarkably useful in discussing more advanced topics. For example,diagonalization theory is discussed via the approximation of powers by a power series of matrices with the operator norm. de Boor also discusses the “localization” of the spectra of a linear map, which allows him define the trace of a linear map and topics such as the the Spectral Mapping Theorum and Gregorian’s Circles, which are rarely covered in beginning courses even at the advanced level. The book finishes with a lengthy and diverse chapter on the applications of linear algebra to geometry, analysis and algebra.  There are so many texts on linear algebra at both the basic and advanced levels one would think it was impossible to write a comprehensive one that didn’t look like the others, one that was geninely creative and offered a fresh perspective on vector spaces. de Boor has written a terrific counterexample to this assumption. But it’s not for dabblers -it’s lucid but quite challenging and requires well-prepared honors undergraduates or graduate students that are committed to a serious mathematics course.  Highly recommended for the right audience.
  22. Advanced Linear Algebra Klaus Kaiser University of Houston (PG) A set of notes for an advanced course at the honors undergraduate or first year graduate level centering on consequences of the fundamental decomposition theorem of linear algebra i.e every vector space is decomposable into a direct sum of cyclic subspaces. The notes go on to cover the spectral decomposition theorem, the Cayley-Hamilton theorem, the Jordan form decomposition, Smith matrix form and symmetric maps and the structure of Euclidean spaces. Not very creative, but very clear,well written and to the point. My one complaint is there are hardly any examples or applications. Still, for a student looking for a concise and clear treatment of advanced linear algebra, this will do very nicely. Recommended.
  23. Linear Algebra James S. Cook Liberty University  Fall 2010  (PG) I’m generally a big fan of Cook’s lecture notes. They combine rigor with intuition and concrete computation better then many other authors’ notes and books-Cook is clearly a passionate teacher who delights in writing his own notes for his courses in order to present his own individual vision for the course. This is another very good example.The notes cover the standard topics: Linear systems and matrices, row and reduced row echelon forms, elementary row  operations, inverse and identity matrices, determinants, the geometry and algebra of vectors in Rand Rvector spaces, linear independence and bases, dimension, row and column spaces, rank and nullity, eigenvalues and eigenspaces, linear transformations and their relationship to matrices, real and complex Euclidean inner products spaces, orthogonal maps and spaces. They are pitched roughly at the same level as Chen’s notes but emphasize applications to geometry somewhat more. They contain both careful  proofs of all major results and many applications to both the physical and social sciences. For example, all the standard mappings of Euclidean 2 and 3 dimensional space are covered in detail with many illustrations, including the inversion map, which usually is studied in complex analysis. There is also Cook’s wonderful writing style and many historical sidebars. Sadly, they’re incomplete and a number of advanced topics which I would have loved to have seen covered-such as multilinear algebra and tensors-are left “blank”. Hopefully future drafts of these notes will develop these topics in detail. Another terrific set of lecture notes by a master teacher. Very highly recommended.
  24. PROBABILITY, STATISTICS AND LINEAR ALGEBRA  C. H. Taubes Harvard University Cambridge Spring, 2010 (PG) An undergraduate level applied course in random processes and linear algebra by the eminent geometer at Harvard who apparently was relaxing by developing this book. Contents: 1 Data Exploration 2 Basic notions from probability theory 3 Conditional probability 4 Linear transformations 5 How matrix products arise 6 Random variables 7 The statistical inverse problem 8 Kernel and image in biology 9 Dimensions and coordinates in a scientific context 10 More about Bayesian statistics 11 Common probability functions 12 P-values 13 Continuous probability functions 14 Hypothesis testing 15 Determinants 16 Eigenvalues in biology 17 More about Markov matrices 18 Markov matrices and complex eigenvalues 19 Symmetric matrices and data sets  The course isn't rigorous, as one wouldn't expect such care in an applied course, as this one clearly is. What it does have are careful definitions, examples and a mind blowingly large set of applications of the material to all manner of subjects-from protein folding to enzyme dynamics to rolling die to tissue growth to bacteria reproduction modeling. And of course, it has Taubes' wonderfully lucid and well organized writing style. Even if one is interested in any of these subjects as a pure mathematics student or teacher, these notes will make a wonderful source of examples and will introduce the reader to entire vistas of applications he or she may have been entirely unaware of. And that makes these notes a terrific asset to everyone. Highly recommended to all as a supplement and to applied students as a text.
  25. Linear Algebra With Applications Oliver Knill Harvard University 2010 (PG) Another version of the same course by Krill-this one with a few applications but nothing more. Again, just good for review.
  26. A First Course in Linear Algebra (A Free  Textbook) by Rob Beezer (PG) Another one of the free online textbooks licensed for use through Creative Commons Licence-they’re springing up like mushrooms now online. Happily, most of them are quite good and some of them are outstanding. Beezer’s book is very much of this new generation of textbooks available for all. It’s exactly what the title says it is-a very standard first course in linear algebra for undergraduates. He covers the standard topics: Linear systems and matrices, row and reduced row echelon forms, elementary row operations, inverse and identity matrices, determinants, the geometry and algebra of vectors in R2 and Rvector spaces, linear independence and bases, dimension, row and column spaces, rank and nullity, eigenvalues and eigenspaces, linear transformations and their relationship to matrices and changes of basis, which Beezer somewhat misleadingly calls representations of vector spaces.  The major innovations here are in the actual implementation and organization of the textbook. Unlike most classical texts, where theorems, definitions and lemmas are numbered, Beezer instead opts for a hyperlink-coded acronym style he attributes to Don Knuth. Each major definition or theorem,instead of being numbered, is referred to in the book by an acronym form which is hyperlinked-allowing readers to click back and forth upon forgetting the statement or proof of the result. For example, the  definition of spanning by a set of vectors of a subspace of a vector space is called  the SSVS property (Spanning Set of a Vector Space) Frankly, I didn’t find this innovation that helpful. It did make the book a little easier to cross reference, but nothing that careful numbering of results wouldn’t accomplish. It may be helpful to students trying to memorize critical definitions and statements of theorems. (Hopefully, they will merely memorize the statements of the theorems and not the proofs themselves, but that’s a discussion for another day. )  The style of the book seeks to encourage a balance between rigorous proofs and concrete computations. Interestingly, Beezer prefers an algebraic approach to a geometric one in a beginning linear algebra course-as a result, there are not many pictures. But the proofs are extremely detailed and the author works quite hard to create many good examples and proof procedures that are easy to understand.  The level of detail of the proofs may annoy hard core math majors and mathematicians that prefer a more concise style, but for students who are encountering proofs for the very first time and/ or learning linear algebra via self study, this book will be enormously helpful. It’s also somewhat conservative in scope, several important topics which are usually covered in such a course are omitted, such as inner product spaces and spectral theory.  However, in a one semester course or self study, this shouldn’t be a problem. Besides, Beezer is currently ( 2015)  working on a follow up book, A Second Course In Linear Algebra,  which will contain all these topics and more-the preliminary version and it’s SAGE source code can be found at http://linear.ups.edu/version3/scla/scla.html (This second book is in a particularly discombobulated state currently and I'll wait for the author to polish it up considerably before I comment on it. )  Beezer  isn’t trying to reinvent the wheel here, but the wheel he’s given us rolls very smoothly, has great craftsmanship and will be quite user friendly for getting students and teachers on the road to learning linear algebra. And of course, the price can’t be beat. Highly recommended for beginners and their teachers.
  27. Fundamentals of Linear Algebra and Optimization  Jean Gallier Department of Computer and Information Science University of Pennsylvania (PG/PG-13)This is a  strange and interesting set of lecture notes from Jean Gallier and his partners in crime at Penn State. Jean Gallier is a completely fascinating and awe-striking person in current academia I'd love to meet someday. These are the lectures from Gallier's course on linear algebra and optimization theory, which forms one of the most active areas of research in numerical mathematics and applied computer science. They give a very clear, complete and very rigorous treatment of linear algebra with many sophisticated examples and applications that are usually not discussed in lecture notes on linear algebra at any level,
  28. such as the finite element method and quadratic optimization. They are pitched at computer science graduate students. As a result, they make no explicit assumptions about their mathematics background beyond an undergraduate discrete mathematics course, but they clearly expect some background in careful mathematics. Gallier writes beautifully and gives many wondeful examples and insights that are not usually given in linear
  29. or abstract algebra courses and precisely defines everything. In short, these notes are a treasure trove for advanced students of pure and applied linear algebra as well as their instructors. They should be mandatory reading for anyone interested in vector spaces. Very highly recommended. Linear Algebra I  Michael Stoll University of Bayreuth Fall 2006 Linear Algebra II  Michael Stoll University of Bayreuth Spring 2007(PG-13) These are the first and second semester lecture notes of a year long course in linear algebra based on Klaus Janich’s very solid textbook. That text is slanted strongly towards the abstract theory of vector spaces and downplays applications. Stoll’s notes are very much in this spirit, but they develop the theory in greater detail with many more examples then Janich’s book. The presentation is fairly discursive without becoming pedantic.They are certainly at a higher level then most U.S. courses in linear algebra, about the same level as Chen or Treil’s notes and perhaps a notch below Erdman’s notes. Stoll develops linear mappings before matricies, which is very in keeping with the purely “algebraic” bent of the notes. The range of the notes is very impressive: Vector spaces and their subspaces, linear transformations, matrices, determinants, orthogonality and inner products, spectral theory (i.e. eigenvalues and eigenvectors), diagonalization theory, dual spaces, tensor products and symmetric mappings and their matrices. The notes are quite modern in spirit; the author introduces commutative diagrams early and uses them throughout-although he stops short of explicitly using category theory. This is a huge positive because it allows him to use the generalizing machinery of category theory, namely commutative diagrams without its subtleties. This allows him to focus on the actual linear algebra without getting sidetracked in Functorland. Stoll writes very well indeed and inserts many personal comments that
  30. are as entertaining as they are enlightening. For example, after a brief digression into how one proves the extension theorem of bases in infinite dimensional spaces using Zorn’s lemma, he concludes with the following: 
    Finally, a historical remark: Zorn’s Lemma was first discovered by
    Kazimierz Kuratowski in 1922 (and rediscovered by Max Zorn about a
    dozen years later),so it is not really appropriately named. In fact,
    when I was a student, one of my professors told us that he talked to
    Zorn at some occasion, who said that he was not at all happy that the
    statement was carrying his name. . .
  31. Stoll makes wonderful digressions like this throughout the notes. My one caveat for them is that, as stated above, they are an uncompromisingly rigorous and comprehensive treatment of the theory of vector spaces and linear maps.They’re definitely not for the faint of heart or the causal student. But for honors undergraduates or graduate students looking to master this critical material, this is one of the best sources you’ll find on the web.  Very highly recommended.
  32. Linear Algebra Jerry L. Kazdan University of Pennsylvania Spring 2013 (PG) A nice set of lecture notes drawing mostly from the problem set collection above. They’re not really intended to act as a full textbook for the course, merely as some additional learning tools. Nicely written and help clarify some basic points, like mappings in Euclidean space. But I still think working through the problem sets with a good set of notes above-like Hefferon or Stoll- will be much more helpful to students as a resource.
  33. Linear Algebra Micheal Taylor University of North Carolina    (PG-13) This is a concise and rapid course in linear algebra extracted from Taylor's recently published textbook on differential equations. Contents: Vector spaces Linear transformations and matrices Basis and dimension Matrix representation of linear transformations Determinants and invertibility Eigenvalues and eigenvectors Generalized eigenvectors and the minimal polynomial Triangular matrices Inner products and norms Norm, trace, and adjoint of a linear transformation  Self-adjoint and skew-adjoint transformations Unitary and orthogonal transformation. Although these notes hit all the important topics of a comprehensive linear algebra course, they're very abstract and concise-they have no pictures and few examples. I doubt they could be used by themselves for anything other then review. That being said-they are extremely careful, well written and have many good exercises. They are particularly good on triangulable matrices and the Jordan form. I'd recommend them for strong students in an honors course or as an intensive review for graduate students. But even so, I'd supplement them with a more applied and/or geometric treatment. Indeed, they'd make an outstanding theoretical supplement to Strang's classic text.
  34. Linear Algebra Eleftherios Gkioulekas University of Texas Pan American (PG) Another wonderful set of notes by Gkiouklekas.Contents: LIN1: Brief introduction to Logic and Sets  LIN2: Brief introduction to Proofs LIN3: Basic Linear Algebra LIN4: Eigenvalues and Eigenvectors LIN5: Vector Spaces LIN6: Vector Spaces – Theory Questions  They aren't as comprehensive as other sources, but Gkiouklekas chooses depth and care over breadth. And I think that's appropriate for a first course,don't you? The notes are exceptionally careful for a first course in linear algebra, beginning with a review of basic logic and naive set theory and building from there, with many wonderful examples and exercises.For example, he gives the elements of group theory before defining vector spaces.  But he also builds up to the general cases gradually, usually introducing concepts in the simpler cases first. For example, he introduces the determinant first for 2 x 2 matrices. Another way he does this is before building abstract vector spaces and linear transformations,he develops matrix algebra fully with many examples and computations. This is how I learned the subject and I think it really makes the most sense no matter how rigorous you ultimately want the course to be.Those computational tools and examples will be needed in order to understand a careful treatment of abstract linear spaces. Can coordinization and the role of bases and their changes under isomorphism be understood by beginners without a good command of matrix algebra? Maybe,but I seriously doubt it. Lastly,he avoids proofs that are either too difficult for beginners or would involve ridiculously lengthy computations, such as a careful proof that det (AB) = det A det B.  Because of this slow building, the student is ready for an abstract approach by the time vector spaces are reached. Each section comes with many examples with full solutions, which are equally divided between computations and model proofs. There are many clever exercises, many with helpful hints and comments. I particularly liked the exercises in the basic logic section, where a series of mathematical sentences are given to the student to convert to logical notation-many drawn from advanced calculus and complex analysis, which the student presumably won't recognize!! This is a set of notes I'd be happy to teach my students linear algebra from. It's one of the best currently available for beginners and I heartily recommend it to all students and teachers of linear algebra. The highest possible recommendation.
  35. Linear Algebra A Second Course FELIX LAZEBNIK University of Delaware (PG-13)These  are a set of notes for a  rigorous course in linear algebra,presuming students already have a lot of experience with matrix manipulation and know the basic elements of vector spaces and their linear maps. Lazebnik writes quite well, but the notes are quite terse and many proofs of theorems are either omitted or only sketched for the students to complete. However, everything is defined very carefully and there are many well-presented examples, both standard and unusual, and these are one of the real strengths of the notes.  The other is the large collection of excellent and diverse exercises, some of them quite challenging.  It's a good set of notes for an advanced student who enjoys working through notes with a pencil in hand or a graduate student who needs to review actively before prelims. But to be honest, all but the best students will find the lack of detail frustrating. For those looking for more detail, there are better sources here, such as Treil,  Gkioulekas or Stoll. Still, Lazebkik's notes will make a good supplement to such a course and is recommended as such.
  36. Linear Algebra and Matrix Theory David Jordan University of Texas Fall 2012 (PG) An excellent set of handwritten notes to accompany a standard introduction to linear algebra. From the course description at the homepage: This course covers a variety of topics in the theory of vectors and matrices and provides an introduction to proofs and abstract mathematics. The course is aimed at students in the mathematical sciences and its objective is to expose students to the basic theory of linear algebra, and to develop their proof-writing skills. The main topics to be covered as time permits are vectors and matrices, systems of linear equations, determinants and eigenvalues, vector spaces, linear transformations and orthogonality. The scanned PDF notes are beautifully written and legible-I'm not sure if Jordan himself or someone else in the class wrote them up, but whoever did did an excellent job. There are many examples, primarily from Euclidean geometry, as well as many pictures. They're relatively short, lucid and very readable and will make a very good text for a short linear algebra
  37. course or self study.The one drawback is that since these notes aren't the primary textbook for the course, there aren't many exercises, so the reader will have to supplement them from elsewhere. Given the enormous number of sources for linear algebra online-most of which are represented here at TULOOMATH-this shouldn't be too hard. Highly recommended.
  38. A Course in Linear Algebra  Matt Kerr University of Washington/UCLA  (PG-13) The title is actually somewhat misleading: this is a preliminary draft of a textbook that Kerr has developed out of an advanced year-long honors linear algebra course he taught at UCLA for several years. So mere mortal students looking for a standard course in linear algebra are advised to look at Kerr’s elementary course, already reviewed above, or Beezer or Hefferon.  It’s actually more of a raw set of lecture notes then an actual online textbook as there are very few exercises, most at the end. It also lacks many computational examples-although the ones it does have are very illuminating and directly motivate theoretical developments. The textbook for the original course was Otto Bretcher’s text-which I think many instructors can testify to the difficulty of using-even in honors courses-for various reasons. (Indeed, profound dissatisfaction with the assigned textbook that the department foists at gunpoint onto the faculty member who’s tasked with teaching the course due to impenetrably labyrinthine university politics is one of the great motivators of lecture note authorship!!!)  We can all be very glad the author not only undertook the task of writing, but posted the notes at his website even after moving on from UCLA. Kerr writes both literately and lucidly, with great care in his definitions and proofs.It’s clearly a course aimed at the very best students in the department, but he doesn’t overdo on the abstraction-he strikes a very good balance between the algebraic and geometric aspects of the subject. He also gives a number of sophisticated applications we don’t normally see in even an honors linear algebra course, such as the Grassmanian on projective space, quarternions, Fourier series, and even atlas constructions on differentiable manifolds(!) For all its flaws, Kerr has written a very substantial, careful and insightful set of notes that will be very helpful for serious mathematics majors without being too abstract or challenging. It will also be terrific supplemental reading for graduate students or instructors of linear algebra. Here’s hoping eventually he combines these notes with the more elementary version to form a standout textbook on linear algebra for both basic and advanced students. Very highly recommended to advanced students.
  39. Linear Algebra Problems Jerry L. Kazdan  University  of Pennsylvania (PG/PG-13)) Another one of Kazdan’s lecture notes, this one a large collection of exercises in linear algebra. These are pitched at a lower overall level then Erdman’s exercise collection described here and the diversity of problem kinds is much higher. What is so impressive besides the large diversity in this collection of exercises is how Kazdan presents the problems on a difficulty gradient: He begins with very simple, intuitive problems and gradually ramps up the level of difficulty.  Towards the end, Kazdan gives very sophisticated problems on the Jordan form and applications of spectral theory. But in between, he gives many wonderful problems on matrices, mappings in Euclidean space, computations with eigenvalues and eigenfunctions, physical applications and much more. This “difficulty gradient” means that the deeper one goes into the problems, the more challenging they become and this is a great asset for
  40. students learning the material or professors using them as a supplement to such a course. Between this and Erdman’s notes, students and teachers looking for free collections of exercises to supplement linear algebra courses at all levels are all set. Very highly recommended.
  41. Discrete Mathematics and Linear Algebra Laszlo Babai University of Chicago 2011 (PG-13) These are a concise set of notes to supplement the University of Chicago’s famous honors undergraduate research “boot camp”, taught by one of the world’s preeminent combinatorialists for many years. The notes give many challenging puzzles and unexpected connections between linear algebra, combinatorics and graph theory. Sadly, most mere mortal undergraduates will merely find them confusing. But I think they’ll be useful for advanced students curious about such matters.
  42. Linear Algebra with Probability Oliver Knill Harvard University Spring 2011 (PG) Another terse set of notes by Krill-these are terser then most, basically just a bullet point outline of a linear algebra course he’s given there. Maybe useful for review, but just too lacking in detail to be useful for anything else.
  43. Linear Algebra Lecture Notes Andrei Antonenko SUNY Stonybrook  (PG)A very nice, careful and surprisingly comprehensive set of lecture notes compiled from the courses Atonenko both attended as a student and later taught as a graduate student in applied mathematics ( doing research in linguistics, of all things!)  The interesting thing about these notes is how they gradually ramp up the rigor level. They begin with a concrete treatment of numbers and naive set theory, then a purely computational treatment of solving linear systems and proceed to discuss matricies over the real field, their operations and inverses. He then proceeds to discuss abstract vector spaces and their associated algebraic properties and important subspaces, such as the nullspace and the kernel. The rigor level then increases steadily and significantly, continuing through linear functions, determinants, Euclidean spaces and orthogonality relations and concluding with a discussion of linear operators in finite dimensional spaces via spectral theory and diagonalization. The notes conclude with a brief but careful discussion of the Jordan form. There’s very little fat here and the author comes right to the point with very little chit chat. However, along the way, he includes many examples, both geometric and algebraic and none are throwaways.  All in all, this is an excellent set of notes, concise and yet very lucid, while containing all the major topics in a serious year long course on linear algebra. Highly recommended.
  44. Linear Algebra by Jim Heffron Saint Micheal's College
    Answers to exercises Linear Algebra Jim Hefferon  (PG)This book’s been getting a lot of attention lately. Naturally, a lot of it has to do with the book being made available freely and officially over the Web by the author via Creative Commons Licence. That’s a shame because even if the book was published via the normal route and it cost’s a life insurance policy vesting for you to be able to buy it, it would still be well worth studying. All the standard topics are covered: Linear systems and matrices, row and reduced row echelon forms, elementary row operations, inverse and identity matrices, determinants, the geometry and algebra of vectors in R2 and Rvector spaces, linear independence and bases, dimension, row and column spaces, rank and nullity, eigenvalues and eigenspaces, linear transformations and their relationship to matrices, real and complex Euclidean inner products spaces, orthogonal maps and spaces. If you’re looking for creativity, then with  one major exception, you’re not going to find it here. What you will find is a standard walk through a first course in linear algebra-but done very thoroughly, skillfully and lucidly. There are many many examples and exercises, as well as insightful remarks by the author in every section. Just about everything is carefully proven and explained very well, with many pictures in low dimensions. Where Hefferon’s text differs quite a bit from the standard linear algebra text in content is in the number and choice of applications. He ends every chapter with a large number of recent and important applications directly based on that chapter’s topic that are usually reserved for more advanced courses-and he presents them very well. For example, at the end of chapter 2 on abstract vector spaces, he gives very interesting discussions of-in order-fields, crystal analysis via construction of a three dimensional cubic vector space, voting theory and the paradoxes and dimensional analysis via linear system analysis. Overall, Hefferon has written a very user friendly, clearly written, mathematically careful and up to date textbook  pitched at the standard undergraduate linear algebra course level that will be of immense use to students and teachers of the subject everywhere. And the fact that it’s free makes that much more attractive. Highly recommended.
  45. Linear Algebra Done Wrong Sergei Treil Department  of Mathematics, Brown University (PG-13)  These
  46. notes’ title is very deceiving. When one talks about linear algebra “done wrong”, one imagines a completely informal treatment with lots of mindless algorithms and computer calculations and no proofs whatsoever-like the stereotype of what an applied linear algebra course is. This is definitely not that kind of course.The “wrong”-ness in the title refers not to lack of rigor, but the choice of topics and how they’re organized, which is quite different from the usual such course. Indeed, it is different from the usual honors course in linear algebra, which this is being pitched at. A standard honors course in linear algebra is generally modeled on Halmos or Hoffman Kunze’s classic textbooks-which is to say they’re usually courses entirely on the raw theory of vector spaces over abstract fields, with very few if any applications. This means they usually take the form of an introduction to abstract algebra. Treil disagrees with this approach and his goal appears to be to provide a fully rigorous course in linear algebra while downplaying the algebraic aspects. In other words, he wants to carefully develop it as a subject in its’ own right. I can certainly sympathize with this approach-it was how I first learned the subject from the late Joseph Hershenov years ago. He does this in several ways. Algebraic structures are not explicitly mentioned except for fields and abstract fields aren’t developed at all-only the real and complex numbers. I  could go into considerable description as to what’s different about this beautiful, inventive and immensely informative text on vector spaces, but the author can speak very well for himself and I think he describes what makes his book unique far better then I can. Therefore,I defer to his own description in the preface. It’s a teacher’s textbook, a book designed to teach linear algebra to the best freshman or more advanced undergraduates.  Several of my own observations: 1) The organization of the book is “telescoping”, What I mean by that is that the author develops the definitions and simplest results of virtually all the general machinery-vector spaces,matricies and linear transformations-of the book in the first chapter and spends the rest of the book “expanding” those topics in detail-sometimes in ways the usual order of coverage makes impossible. For example, by developing matrix algebra and linear transformations in the first chapter, this allows the author to develop solving linear systems in the second chapter as linear transformations on an m × n matrix that result in row echlon or reduced row echlon form where the solution space is obvious by inspection-and these transformations are expressible as multiplication of the matrix by elementary matricies. This is usually done by abstract linear transformations in advanced courses, but combining the 2 approaches- concrete computation algorithms and abstract linear transformations-makes the whole presentation so much more unified and clearer. It also allows him to emphasize the role of pivot rows in this conversion-which leads directly to the relationship between reduced row echlon form of an real valued m x n  matrix and a basis of Rn. 2) The theory of determinants-with the exception of the Jordan normal form-is probably the hardest concept to discuss in a theoretical linear algebra course. Treil does an outstanding job by first developing the geometry and calculational aspects of the determinant first and then defines it rigorously via real valued permutation mappings. He later returns to determinants in the chapter on multilinear algebra. To my knowledge, with the possible exception of Charles Curtis’ book, this is the most complete and coherent treatment of determinants at this level I’ve ever seen. 3) The discussion of the dual space is considerably more thorough then one usually sees in advanced courses in linear algebra. The author stresses the various aspects of the relationship between a vector space and it’s dual and the chapter contains a finite dimensional proof of the Riez representation theorem, which got me to hello on that. More importantly, the vector space/dual space discussion leads to one of the best introductions to tensors and multilinear algebra I’ve ever seen via differential operators. Physics students who are suffering trying to learn this material will find Treil’s introduction to the Einstien convention a gift. All in all, I think Triel has succeeded marvelously at writing an original yet challenging linear algebra book for strong students, one that will certainly inspire them to go further in mathematics. I would strongly advise the author to continue to make this book available online via Creative Commons License or to self publish it to keep the cost down. If he does-this book may very well replace the old classics as the standard advanced course on the subject. As it is, I’m doing my best to get the word out on it. Very  highly recommended.
  47. Basics of Linear Algebra Fedor Duzhin Nanyang Technological University (PG)  This is a set of notes on both analytic geometry and basic linear algebra to lay the foundations for Duzhin's rigorous course in calculus at several variables at Nanyang, which we already commented on here.Contents:1 Polar coordinates 2 Gaussian elimination 3 Matrix multiplication and finding inverse matrix 4 Determinant 5 Determinant Continued 6 Row and Column Expansion 7 Concept of Dimension 8 Vector Spaces 9 Basis 10 Linear Dependence 11 Rank 12 Rank, Determinant, and  Matrix Multiplication The notes are suprisingly sophisticated and careful, as one would expect such a set of notes to focus on computational aspects of the subject. It's clear, however, that these notes are also intended to serve as the basis-no pun intended-of a careful first course in linear algebra. While it does contain quite a bit of plug and chug matrix manipulation and determinants, it also contains rather careful developments of these subjects and much more. For example, the determinant is defined as a real valued function of an n × n matrix argument such that (a) it is multi-linear; (b) it is skew-symmetric; (c) its value on the identity matrix is 1. The notes have a particularly lucid development of linear independence and dimension I liked immensely.There are also many, many terrific examples. Unfortunately-and it's a big drawback- Duzhin omits any exercises from these notes. It's a real shame because that really limits their usefulness.Still, supplemented with appropriate exercises, these notes would make an excellent and sophisticated first course in linear algebra. Highly recommended as a supplement.
  48. Fundamentals of Linear Algebra Marcel B. Finan Arkansas Tech University (PG) Yet another excellent set of lecture notes from Finan that form the foundation for a very solid first course in linear algebra. Contents: 1 Linear Systems 2 Matrices 3 Determinants 4 The Theory of Vector Spaces 5 Eigenvalues and Diagonalization 6 Linear Transformations This is a purely theoretical course pitched at the undergraduate level, so it doesn't go too deeply or abstractly into the theory of linear spaces.But what it does cover,it covers superbly. Like all of Finan's online texts, they are supremely organized, clearly and simply written,they have many good examples and solved problems and a truckload of exercises for the student to chew on. Best of all, complete solutions to the exercises can be found here. Finan believes all lecture notes and textbooks should be suitable for self study by students with out a teacher-a wonderful philosophy all his educational works are good examples of and this one is no exception. Finan begins with linear systems of m equations in n unknowns and develops the general theory of linear algebra slowly from there-the contents show the very natural progression from the concrete and specific to the abstract and general. Linear equations precede matrices and thier computations, which leads to the determinant as both a calculational tool and a theoretical construct, emphasizing the sign function and elementary matrix decompositions. Only once all this computational machinery and the corresponding truckload of examples and beginning experience with careful definitions and theorems is mastered by the student does the author then proceed to abstract vector spaces and linear mappings. This is how I learned linear algebra as an undergraduate as my first "real" mathematics course. I still think there's really no more effective way for students with no prior experience with careful mathematics to learn it. One should be warned that despite it's elementary level, this is a purely mathematical treatment of vector spaces and if one's looking for applications, you need to look elsewhere. But for budding mathematics majors who are trying to begin to understand rigor, this is one of the very best online sources from which to learn elementary linear algebra-and applications can easily be supplied from other sources.Highly recommended to all beginning students and teachers of linear algebra.
  49. Introductory Notes in Linear Algebra for the Engineers Marcel B. Finan Arkansas Tech University  (PG) These notes were extracted from the more theoretical version of the linear algebra course Finan developed and commented on here. This is one is specifically aimed for the needs of engineering  students. As one would expected, there is considerable overlap between them and the more mathematical version. Also as expected, many of the theoretical discussions and proofs have been omitted and it is less careful and much more practically and computationally minded.It focuses much more on matrix reduction algorithms and specific applications of importance in geometry and the physical sciences,such use of eigenvalues and eigenvectors to solve systems of linear ordinary differential equations. There are, like it's predecessor, a ton of excellent problems and complete solutions to them for this version can be found here.To be honest, I really don't think ripping the proofs out of a set of linear algebra notes and passing off what remains as a course for engineers is really doing them any favors-the theory of abstract vector spaces is now critical in the physical sciences, especially quantum theory and it's applications to engineering.Be that as it may, if you're  looking for a somewhat gentler and more applied version of Finan's  lectures that still possessed the wonderful readability and scholarship  of the author, here it is. Be my guest.
  50. Dr. Z.'s Introduction to Linear Algebra Posted Solutions   (PG)This are full solutions to the various quizzes and exams Doron Zeilberger gave in his linear algebra course at Rutgers University over the years. They show excellent taste-a very good combination of rigorous proof and computational exercises. Very good supplemental practice for linear algebra students, highly recommended as such.
  51. Algebra I- Linear Algebra Ulrike Tillmann  University of Oxford 2014   (PG)An excellent set of handwritten notes and problem sheets in the Oxford style for a rigorous first course in linear algebra in the UK style.  At Oxford, this is the first course in the core "undergraduate" 3 semester abstract algebra sequence-the prerequisite is  plug and chug matrix algebra course usually given at the "prelim" level that covers the basics of linear equations,matrices and linear mappings without vector spaces. Contents:  Definition of a vector space over a field (expected examples ). Examples of vector spaces: solution space of homogeneous system of equations and differential equations; function spaces; polynomials;  as an -vector space; sequence spaces. Suspaces, spanning sets and spans. (Emphasis on concrete examples, with deduction of properties from axioms set as problems). Linear independence, definition of a basis, examples. Steinitz exchange lemma, and definition of dimension. Coordinates associated with a basis. Algorithms involving finding a basis of a  subspace with elementary row operations (EROs). Sums, intersections and direct sums of subspaces. Dimension formula.Linear transformations: definition and examples including projections. Kernel and image, rank nullity formula. Algebra of linear transformations. Inverses. Matrix of a linear transformation with respect to a basis. Algebra of matrices. Transformation of a matrix under change of basis. Determining an inverse with EROs. Column space, column rank. Bilinear forms. Positive definitesymmetric bilinear forms. Inner Product Spaces. Examples:  with dot product, function spaces. Comment on (positive definite) Hermitian forms. Cauchy-Schwarz inequality. Distance and angle. Transpose of a matrix. Orthogonal matrices.These notes assume the student knows and understands all the practical aspects of the subject and jumps directly into a fully abstract presentation of the subject, with many examples, mostly from the number systems and geometry. If you're looking for a course to teach you how to row reduce an n x n matrix-look elsewhere. For a rigorous course that's very clear and relatively brief, this would be a good pick for either strong students to self study or teachers of linear algebra. It's not the best-either Baker or Cameron are better for this kind of course.But not bad at all. Recommended.
  52. Linear Algebra II Alan Lauder University of Oxford February 14, 2014 (PG-13)A sophisticated and very concise (23 pages!), yet shockingly readable set of notes and exercise sheets for a second undergraduate course in linear algebra at Oxford. Contents: Introduction to determinant of a square matrix: existence and uniqueness and relation to volume. Proof of existence by induction. Basic properties, computation by row operations.Determinants and linear transformations: multiplicativity of the determinant, definition of the determinant of a linear transformation. Invertibility and the determinant. Permutation matrices and explicit formula for the determinant deduced from properties of determinant. Eigenvectors and eigenvalues, the characteristic polynomial. Trace. Proof that eigenspaces form a direct sum. Examples. Discussion of diagonalisation. Geometric and algebraic multiplicity of eigenvalues.Gram-Schmidt procedure.Spectral theorem for real symmetric matrices. Matrix realisation of bilinear maps given a basis and application to orthogonal transformation of quadrics into normal form. Statement of classification of orthogonal transformations. And yes, the author does indeed get all this material in in 23 pages, which gives you an idea what they're like..Amazingly,as I said, the notes are quite readable with good examples.The author achieves this by being extremely selective about what he covers-and then focusing on it.As a result, a great deal of important material is covered here rigorously if concisely.A complete and careful development of the determinant and the spectral theorem are the real highlights of these notes. A very good set of notes for a follow-up course in linear algebra for strong undergraduates. Highly recommended!
  53. Linear Algebra I Ulrich Meierfrankenfeld  Michigan State University Fall 2011 
  54. LinearAlgebra II  Ulrich Meierfrankenfeld Michigan State University November 10, 2001   (PG-13)More then a decade apart  from each other, these are Meierfrankenfeld's lecture notes for the 2 semesters of linear algebra offered to undergraduates at Michigan State. They are quite well written,careful and show a good choice of topics-together, they can certainly form the nucleus (Ah,thought I was gonna say basis again,huh?Nyah nyah again!) for a year long course aimed at mathematics majors. Contents (2011 Course I)  1 Vector Spaces 2 Systems of Linear Equations 3 Dimension Theory 6 Linearity 5 Matrices 6 Linearity (Cont.) 7 Determinants 8 Eigenvalues and Eigenvectors A Functions and Relations B Logic C The real numbers D General Commutative and Associative Laws (2001 Course II)1  Introduction 2 Fields 3 Polynomials and the field of rational function 4 Direct sums and linear independency 5 Bases and Dimensions 6 Quotient Spaces And The Isomorphism Theorem 7 Polynomials, the division algorithm and algebraically closed fi eld 8 Eigenspaces 9 Hermitian Forms 10 The dual space, adjoint operators and the Spectral Theorem  The intersection of the 2 lecture notes is not empty-the overlap occurs mostly in the sections on linear independence and bases of vector spaces. However, the discussion in the 2001 notes is considerably more sophisticated, as one would expect. I have 2 minor carps about these notes. My first is that there are no applications. That's really a matter of taste, though-if Meierfrankenfeld wants to present linear algebra as pure mathematics, that's his prerogative. But it's kind of like presenting calculus without any applications-it only tells half the story and gives the impression the subject is half as important as it actually is. My other complaint with both these notes-and it's particularly a problem in the 2001 notes-is that there are relatively few examples and the ones Meierfrankenfeld does develop he tends to use over and over again. For example, he tends to use the set of all m ? n invertible matrices as his example of a vector space of matrices and its subspaces. Not that that's bad, mind you-but some diversity in exercises always makes a set of notes a more effective educational tool. The real strength of these notes is Meierfrankenfeld's enormous, almost anal care with precise definitions and theorem statements, something you don't normally see in courses at this level.  For example, he actually gives a precise definition of an m x n real matrix. That may seem like a little thing, but ask yourself: How many of even the most careful treatments of vector spaces actually give one? Overall, an excellent mathematical treatment of linear algebra and will be a very good asset for both students and teachers. Highly recommended.
  55. Basic Linear Algebra Andrew Baker University of Glasgow 2009    (PG) Beautifully written, superbly organized and deep set of lecture notes for a first year UK style university course on linear algebra that's one of my favorites of the currently available online lecture notes. Contents Chapter 1. Vector spaces and subspaces  1.1. Fields of scalars 1.2. Vector spaces and subspaces Chapter 2. Spanning   sequences, linear independence and bases 2.1. Linear combinations and spanning sequences 2.2. Linear independence and bases 2.3. Coordinates with respect to bases 2.4. Sums of subspaces Chapter 3. Linear transformations 3.1. Functions 3.2. Linear transformations 3.3. Working with bases and coordinates 3.4. Application to matrices and systems of linear equations 3.5. Geometric linear transformations Chapter 4. Determinants 4.1. Definition and properties of determinants 4.2. Determinants of linear transformations 4.3. Characteristic polynomials and the Cayley-Hamilton theorem Chapter 5. Eigenvalues and eigenvectors 5.1. Eigenvalues and eigenvectors for matrices 5.2. Some useful facts about roots of polynomials 5.3. Eigenspaces and multiplicity of eigenvalues 5.4. Diagonalisability of square matrices Appendix A. Complex solutions of linear ordinary differential equations Appendix. Bibliography I love all of Baker's online course notes, which he's been very generous in making freely available at his website. He is a master at choosing and organizing definitions, examples, theorems and proofs in his lectures so that there's virtually no extraneous fluff-everything in the notes is of great significance for the given topic. Or as Guy Fieri likes to say on his show, all thriller, no filler. This allows him to cover each topic in great depth in a relatively small number of pages. The result is a treatment of linear algebra that's deeper in some ways then books or notes that are 5 times it's length. Quality not quantity. The discussions of diagonalization and the Cayley-Hamilton theorem are exceptional.Print out a copy of these for yourself or your students if you're either taking or teaching an introductory linear algebra course. You'll thank me later. The highest possible recommendation.
  56. Linear Algebra David Lerner Department of Mathematics University of Kansas   (PG-13)A very strong and comprehensive set of lecture notes that Lerner composed for the honors course in linear algebra at the University of Kansas.Contents 1 Matrices and matrix algebra 2 Matrices and systems of linear equations 3 Elementary row operations and their corresponding matrices 4 Elementary matrices, continued 5 Homogeneous systems 6 The Inhomogeneous system Ax = y, y 6= 0 7 Square matrices, inverses and related matters 8 Square matrices continued: Determinants 9 The derivative as a matrix 10 Subspaces 11 Linearly dependent and independent sets 12 Basis and dimension of subspaces 13 The rank-nullity (dimension) theorem 14 Change of basis 15 Matrices and Linear transformations 16 Eigenvalues and eigenvectors 17 Inner products 18 Orthonormal bases and related matters 19 Orthogonal projections and orthogonal matrices 20 Projections onto subspaces and the Gram-Schmidt algorithm 21 Symmetric and skew-symmetric matrices 22 Approximations - the method of least squares 23 Least squares approximations - II 24 Appendix: Mathematical implications and notation The author's prefaces to this work is well worth reading as it is very much in the spirit and intent of my intent of crafting this archive and it's accompanying resources.To quote the author's preface: These are notes, and not a textbook; they correspond quite closely to what is actually said and discussed in class. The intention is for you to use them instead of an expensive textbook.......       How can any serious student or teacher not immediately,out of sheer gratitude,at least look at them? If he or she does read through them, they'll be quite impressed. I know I was.Lerner makes several choices that allow him to present linear algebra rigorously yet concretely. His major decision is to insist all vector spaces and their subspaces are constructed on R.Therefore, all matrices have real entries. Abstract or infinite dimensional vector spaces and their isomorphisms are not discussed at all. This will upset a lot of purists, but it has the enormous advantage for beginners of ensuring all examples of vector spaces and computations with matrices and linear maps are straightforward and clear to the students.The notes are also structured differently then one would expect a course like this to be. The first 10 chapters of the notes are very informal, with many calculations and examples and few theorems and proofs-the level of rigor and careful proof very gradually ramps up.In chapter 10, on subspaces, the rigor takes a huge leap up and the course begins to look much more like a traditional theoretical treatment, but this is only after a large number of examples and calculation techniques are mastered.All these tools are brought to bear in the motivation and proofs of major theorems.To give just one example, Chapter 5 is entirely about computational solutions to homogeneous systems-in chapter 10, one of the first examples of a subspace is the solution set of Ax = 0, which immediately gives the definition of the null space! It is almost as if nearly all the rigor is separated out into the second half of the course where all the calculations are generalized by theorems. This is a strikingly original approach, one I think most students will like immensely once they "get it". The author is very adamant about the role of definitions in mathematics and this approach has the added benefit of allowing the definitions to be "extracted" easily from the earlier computational material. The author has a very personable and clear style and seems to know exactly where he's going with this at all times. There are many exercises, most not difficult and  the difficult ones are starred as a warning. I have just one minor quibble-there aren't many applications to other areas of mathematics or the physical or social sciences in these notes. But this is a very minor
  57. quibble. Lerner has written one of the most comprehensive and illuminating introductions to vector spaces currently available for download. I hope he continues to polish it and above all, continue to make it available freely to both students and teachers of linear algebra. Most highly recommended!
  58. Linear Algebra Course Fall 2010 Peter Dodds University of Vermont (PG) This is a rather erratically organized set of supplementary lecture material for an applied linear algebra course based on  Introduction to Linear Algebra by Gilbert Strang, which is a vastly diminished and inferior "introductory" version of Strang's incredible Linear Algebra With Applications. This is rather ironic, since that's more or less how Dodds' notes strike me-inferior and cut-down versions of the real course material, which is really contained in the textbook. Some nice humorous insights on applications and linear systems, but really not a whole lot here that can't be found done well in the textbook. You can even find the material done better in Strang's streaming lectures at MIT, which of course are freely available through OpenCourseWare here. I'd pass.
  59. Linear Algebra II Dave Penneys University of California Berkeley August 7, 2008  (PG-13) A very strong abstract presentation of linear algebra that goes surprisingly far for an undergraduate course without using advanced material from algebra or analysis. It does presume  a good working knowledge of the manipulative aspects of linear algebra such as matrix arithmetic and Gaussian elimination as well as some knowledge of the basic definitions of vector spaces and  linear maps. It also has several strikingly original touches that distinguish it from other such sources online. Contents 1 Background Material 2 Vector Spaces 3 Linear Transformations 4 Polynomials 5 Eigenvalues, Eigenvectors, and the Spectrum 6 Operator Decompositions 7 Canonical Forms 8 Sesquilinear Forms and Inner Product Spaces 9 Operators on Hilbert Space 10 The Spectral Theorems and the Functional Calculus The first and most striking original touch in the presentation is that Penneys while he gives a multitude of examples of the defined objects and proved theorems, he states them without explanation or proof. The result is that the reader is given the very helpful task of constantly verifying these examples,which are usually very straightforward tasks.This active learning of examples has the side benefit of inserting a very large number of simple exercises directly into the text without the frustrating practice of inserting exercises where major results that are needed for later topics have to be proven by the student A lot of mathematics textbook authors and teachers struggle with the corundum of how much of the material to leave for students to prove without making it self defeating for them-this is a brilliant solution to the problem and I wish I'd thought of it.Another original aspect is the strong emphasis Penneys puts on eigenspace theory.The discussion of canonical forms and the Jordan form is particularly good (we can never have too many of those!) Lastly, he gives a very modern presentation of orthogonal linear operators and their inner product spaces via as finite dimensional Hilbert spaces. This allows him to state and prove the finite dimensional analogues of operator theory in modern language,such as Riez representations and unitary maps, while still maintaining an elementary level without topology or integration theory. The result is that students that go on to graduate school will have a basic grasp of  the vocabulary and not have to completely relearn it.I was really surprised how much I liked these notes. They are challenging but quite well written and organized. The self-studying student, who likes to read with pen and paper in hand, will particularly benefit from these excellent notes.But any student who puts in the effort of reading and working through them will be rewarded with a deep understanding of  linear spaces. Highly recommended to all students and teachers of linear algebra
  60. Linear Algebra Aravind Asok University of Southern California Spring 2012  (PG-13) Concise but well organized and readable set of lecture notes and exercises for a second course in linear algebra.The book for the course was Kenneth Hoffman and Ray Kunze's classic Linear Algebra,which is still considered by many to be the definitive theoretical treatment of linear algebra. In it's heyday in the halycon 1970's, strong institutions like MIT and Harvard used it as an uncompromisingly abstract and complete introduction to vector spaces.Even if your students are strong enough to handle it, the big problem with this kind of high-tech approach to a beginning course in linear algebra is the same problem you have with a purely theoretical treatment of calculus for honors freshmen-it basically only tells half the story.The applications of linear maps and their computational methods are at least as important in both pure and applied mathematics as the raw theory of linear spaces.In any event, these aren't intended to replace the course text, only supplement it with added proofs, computational examples and exposition that the brute conciseness of the text omits.They are surprisingly substantial considering they are not a course text. The overriding theme of Asok's notes is how to motivate the solution of the general eigenvalue problem using the theoretical structure behind the totality of computational techniques students already know before taking this course. Asok gives many nice insights that help illuminate the dense discussions in Hoffman/Kunze- particularly in giving a very detailed and careful treatment of the determinant function as well a full proof of the Cayley-Hamilton Theorem .The homework problems are equally meaty and challenging, involving both proofs and computations.I like them, but to be honest, there's really nothing here that can't be found done as well or better in Treil,  Gkioulekas, Meierfrankenfeld, Baker, Stoll or a half a dozen other sources here. Still, I'd recommend them as a supplement to a  serious linear algebra course because I like Asok's style and didactics. Recommended.
  61. Abstract Linear Algebra  Noel Brady University of Oaklahoma August, 2000 (PG)Very concise and dry set of lecture notes for a second course in linear algebra. They're readable, but far too brief in many places-in some places,such as in the chapter on canonical forms,  they're flat out incomplete!  They're similar in approach and content to Penneys' notes above, but they completely lack Penneys' breadth, superior organization and selective expository bursts that make those notes such a pleasure to read. .Contents Part 1: Basics Vector spaces, Linear transformations, Matrices,Dual spaces, Determinants.Part 2: Structure of Linear Transformation Space  Facts from algebra of polynomials, eigenvectors/values, characteristic polynomials, primary decomposition, rational and Jordan canonical forms Part 3: Inner products Inner product spaces, dual spaces revisited, self-adjoint, unitary and normal operators, spectral theory, structure of bilinear forms,Sylvesters law.Part 4: Miscellaneous topics Linear groups and geometry. Brady gives good exercises to chew on and he's got a good sense of humor that makes some passages amusing (for example, referring a property of determinants as the source of "the hateful exercises on the determinant" in the introductory course) They're ok for all their flaws, but they didn't really impress me and the sloppy omissions annoyed me. I think you're better off with Penneys or Stoll for such a course.
  62. Advanced Linear Algebra I Vaughn Climenhaga University of Houston 2013 : (PG-13) A substantial and interesting set of lecture notes for a second course in linear algebra based on the advanced text by Peter Lax. What makes Lax different from most of the other advanced linear algebra texts is that it focuses on the analytic aspects of vector spaces i.e. the theory of linear operators in finite dimensional spaces. The book is really intended as a prelude to his functional analysis text, to review and strengthen students' knowledge of the specific aspects of linear algebra needed before studying that book. Climenhaga's notes are complete unto themselves as a text for this course. They're a bit more balanced-less analytic and more algebraic then Lax's book, as well as being more detailed-while covering much of the same material at at the same level.  Contents: Vector spaces and linear maps from the abstract point of view; determinant, trace, and spectral theory (eigenvalues, eigenvectors) of linear maps; and the structure of Euclidean space.The notes are literate, lucid and quite broad, covering a lot of additional ground to what's covered in the first 7 chapters of Lax. In short, they're a pleasure to read. Matrices are downplayed in favor of linear operators, as is appropriate for a course set on abstract vector spaces.For example, he freely introduces amd  uses commutative diagrams to illustrate isomorphism theorems.The use of commutative diagrams in linear algebra is very straightforward and clarifying-additionally, it gives good practice for the study of module theory over general rings. He also gives a particularly long and detailed development of the determinant function, one of the best I've seen. There's also a terrific chapter on the linear algebra of Markov chains and their manifold applications, including chemical kinetics and the Google pagesearch algorithm. There are many examples, all clearly presented with details as well as many good meaty exercises to supplement the ones in Lax.Overall, this is an excellent set of notes for a second course in linear algebra and will serve both students and teachers well in this capacity. Very highly recommended.
  63. Linear Algebra and Differential Equations Worksheets Course Materials And Scanned Lecture University of Berkeley Math 54 7 th Edition 2012 edition   (PG)These are a large set of supplementary problem sets for the undergraduate combined first course in linear algebra and differential equations at Berkeley. Contents 1. Introduction to Linear Systems 2. Matrices and Gaussian Elimination 3. The Algebra of Matrices 4. Inverses and Elementary Matrices 5. Transposes and Symmetry 6. Vectors 7. General Vector Spaces 8. Subspaces, Span, and Nullspaces 9. Linear Independence 10. Basis and Dimension 11. Fundamental Subspaces and Rank 12. Error Correcting Codes 13. Linear Transformations 14. Inner Products and Least Squares 15. Orthonormal Bases 16. Determinants 17. Eigenvalues and Eigenvectors 18. Diagonalization 19. Symmetric Matrices Differential Equations 20. The Wronskian and Linear Independence 21. Higher Order Linear ODEs 22. Homogeneous Linear ODEs 23. Systems of First Order Linear Equations 24. Systems of First Order Equations–Continued 25. Oscillations of Shock Absorbers 26. Introduction to Partial Differential Equations 27. Partial Differential Equations and Fourier Series 28. Applications of Partial Differential Equations It's mostly a computational course, although it does gives the barest elements of the theory of the solution of linear equations and the basics of differential equations. They aren't really a course text and don't claim to be-they're a diverse collection of challenging exercises covering the terrain of this course and providing students with additional training and experience in more difficult computations and solution methods then the textbook exercises provide. A large number of Berkeley faculty contributed to this collection and their names are given in the preface.The problems are well thought out and clear, challenging without being too much so and will give much food for thought while the students work through them. For strong students and teachers, this collection gives an excellent,diverse and very handy collection of terrific exercises to work through and assign.Many of them test practical problem solving in the physical sciences, particularly engineering.To give an idea as to the range of the problems, some topics used: Kirchikoff's laws of current flow,geometric transformations in the plane, error corrective codes, ordinary and partial differential equations and thier methods of solution and Fourier series. This is a very valuable resource and everyone either taking or teaching elementary linear algebra or differential equations should bookmark them on their
  64. computer. Very highly recommended!
  65. Linear Systems Joel Feldman University of British Columbia Spring 2011  (PG)These are Feldman's excellent supplementary notes to a first linear algebra course based on the equally terrific online textbook/lecture notes by his UBC colleagues Richard Froese and Brian Wetton we already commented on here. Contents: vectors and coordinate representation; vector length, dot product,
  66. projection,determinants; cross product; lines in 2D and 3D and planes in 3D;geometry of solutions of linear systems; linear dependence and independence; solving linear systems; solving linear systems (cont.); echelon form and rank; homogeneous equations;resistor networks;resistor networks (cont.); matrix multiplication; linear transformations; rotations, projections and reflections in 2D;matrix representation and composition of linear transformations;random walks; transpose;matrix inverse; matrix representation of resistor network problems;determinants; determinants (cont.); complex numbers; complex linear systems;eigenvalues and eigenvectors; eigenvalues and eigenvectors (cont.); powers of a matrix;application of eigen-analysis to random walks;vector differential equations; application of vector DEs to electrical networks. The result is he succeeds in making a very good source on elementary linear algebra even better by putting his additional 2 cents in. The notes have Feldman's usual wonderful writing style, depth, intuition and clarity and build further on the topics in Froese/ Wetton. Like Freose/Wetton, the notes develops concrete vector algebra in R2 and R3 in it's entirety with many applications to both geometry and physics concurrently with the elements of abstract vector spaces. This is an immensely fruitful approach for a first course and allows subsequent abstract courses to build on the resulting deep intuitive understanding. It also allows the coverage of many important applications without loss of rigor. There are many examples, good problems of varying difficulty, careful proofs and geometric diagrams illustrating the concepts. His main additions to the Froese/Wetton "text" are his extensions of their discussions to complex vector spaces and matrices and several additional applications, mostly to geometry and electrical network theory.  The result is that the union of these notes and Froese/Wetton form one of the very best introductions to linear algebra that exists either in print or online and every student and teacher needs to booklink to it. They'll be using it on an almost daily basis. The highest possible recommendation!
  67. CHANGE OF BASIS AND ALL OF THAT LANCE D. DRAGER Texas Tech University   (PG) Brief, readable notes on change of basis in linear algebra for a basic course culminating in the diagonalization formula for basis transformations. Not bad, but really nothing you can't find in one of the more substantial sources. You can check it out, but nothing major here.
  68. Linear Algebra Daan Krammer and Agelos Georgakopoulos University of Warwick March 2013   (PG)A rigorous and concise, but quite readable first course in linear algebra. It is not as comprehensive as some of the other sources here, so it can be covered in a semester or so by either a class or by self study.Contents: 1 Number systems and fields 2 Vector spaces 3 Linear independence, spanning and bases of vector spaces 4 Subspaces 5 Linear transformations 6 Matrices 7 Linear transformations and matrices 8 Elementary operations and the rank of a matrix 9 The inverse of a linear transformation and of a matrix 10 The determinant of a matrix 11 Change of basis and equivalent matrices 12 Similar matrices, eigenvectors and eigenvalues The material is fairly standard- presented carefully and no extraneous chit chat,but with a good number of detailed examples as not to become a bullet point presentation. This is a typical careful linear algebra course in the UK style-well written and organized, but probably too difficult for any but an honors first course in the US.A good set of notes, nice and readable and relatively brief. The big problem with these notes is that the exercise sets appear to be unavailable outside of the University's Moodle system at Georgakopoulos' webpage.You'd have to supplement them with exercises, which is a huge drawback. But if you're looking for a presentation that's careful and to the point without any sidebars and don't mind the lack of exercises,this is a good choice. Recommended
  69. .Linear Algebra Supplementary Notes Reyer Sjamaar Cornell University    (PG-13)   Excellent supplementary notes for an honors course in linear algebra at Cornell. Contents:Chapter 1 Fields And Vector Spaces Chapter 2 Determinants Chapter 3 The Jordan Normal Form Chapter 4 Multilinear Algebra Appendix A Bases And Dimension The central theme of these notes is to develop the machinery needed for tensors, differential forms and multilinear algebra. This means primarily dual spaces, tensor product and the Jordan form and their associated concepts.  In particular,Sjamaar gives one of the best presentations of the Jordan form and generalized eigenvalues I've ever seen. I wish I'd had them when I was first learning this material. The notes are immensely readable and informative,rigorous without being dry and not too pedantic. There are also many examples and good exercises. Whether or not the ulterior motive of the author is to provide a theoretical foundation for his equally excellent undergraduate notes on differential forms is a good question.In any event, they certainly can act in this role and the pair of notes together can certainly serves as a text for a full second semester of linear algebra or an undergraduate seminar on multilinear algebra with it's applications. An excellent source on advanced linear algebra for students with a semester of linear algebra. Highly recommended. Linear Algebra I Shmuel Friedland University of Illinois Chicago  Spring 2013 (PG) A standard but very thorough and readable set of lecture notes for a careful first course in linear algebra. Contents: Matrices, Gaussian elimination, vector spaces, linear transformations, orthogonality, Gram-Schmidt process, determinants, inner ucts, eigenvalue problems, and an introduction to Jordan canonical form. Particularly impressive are the historical profiles of the lesser known major players in the development of the subject, such as Gram, Schmidt and Schwatz and the additional important topics that one usually finds in more advanced courses, such as the matrix exponential,which is so critical in multivariable analysis and differential equations.There are many very good examples and solved computations. Sadly, there are very few exercises. but again, these are easily supplied from elsewhere. Highly recommended as a first course alongside a collection of exercises.
  70. Linear Algebra  Jeffrey M. Lee  Texas Tech University Spring 2013   (PG)Lee's PDF converted PowerPoint notes for his undergraduate linear algebra course, which are currently posted at the course webpage.The notes for only about half the course are posted, which is a shame. Powerpoint notes are always expanded enormously when converted to PDF, which makes them more accessible amd readable on the internet but the down side is it results in gigantic files. He covers the standard material for such as course but he Contents:  Lect.1 The Geometry of Linear Equations Lect. 2 Some Key Ideas  Lect. 3 Multiplication and Inverse Matrices, Transposes, Permutations  Lect. 4 Gaussian Elimination, Elementary Matrices  Lect. 5 LU Factorization Lect. 6 Vector Spaces, Subspaces, Span 3  Lect. 7 Column Space and Nullspace Lect. 8 Solving Ax = 0: Pivot Variables  Lect. 9 Solving Ax = b: Row Reduced Form of a Matrix Lect. 10 Independence, Basis, and Dimension Lect. 11 The Four Fundamental Subspaces, The Fredholm lternative Loose ends Change of basis Lect. 12 Matrix Spaces; Rank 1; Small World Graphs Lect. 13 Graphs, Networks, Incidence Lect. 14 Change of basis. More on Abstract Vector Spaces and Linear Transformations, Isomorphism Lect 14* Special Evening Session: Linear Transformations, Matrix and Integral Representations. Lect. 15 Introduction to Inner products. Orthogonal Vectors and Subspaces The course is based on Ron Larson's text, so I'd expect them to be pretty standard. And for the most part they are. Unfortunately,there are some unusual and deep topics covered in the actual lectures that make very welcome  additions to an undergraduate course, such as graphs, incidence matricies and matrix spaces-and none of these notes are posted at Lee's website. Which is aggravating. Still, Lee's notes are quite lively and clear and will make a very nice supplement to any linear algebra course. Recommended.
  71. Linear Algebra Micheal Hill University of Virginia Spring 2008  (PG)An extensive set of handwritten notes for an undergraduate course in linear  algebra complete with exercise sets. Contents:  Lecture 1: Systems and Matrices  Lecture 2: Gaussian Elimination  Lecture 3: The Vector Space R^n  Lecture 4: Span, Independence, Basis, and Dot Product Lecture 5: Geometry and Applications Lecture 6: Matrix Operations  Lecture 7: More Matrix Operations Lecture 8: Inverse Matrices & Matrix Transformations Lecture 9: LU Decomposition Lecture 10: Matrix Operations & Linear Transformations  Lecture 11: Applications Lecture 12: Determinants Lecture 13: Determinants & Inverse Matrices  Lecture 16: Abstract Vector Spaces Lecture 17: Linear Independence & Basis Lecture 18: Basis & Rank Lecture 19: Projections and Gram-Schmidt   Lecture 20: Linear Transformations, Kernel & Range  Lecture 21: 1-1 and onto, Invertible transformations, and Systems Lecture 22: Quadratic Forms Lecture 23: Coordinate Vectors  Lecture 24: Matrices and Linear Transformations Lecture 25: Inner Products & Fourier Series Lecture 26: Applications and Least Squares  The notes are pretty standard in content, Hill isn't trying to reinvent the wheel here. But he does make a very sturdy and well shaped wheel that will get his students well on their way to an understanding of linear algebra. (Yeah, I know-but I like that metaphor and it's my site,so nyah nyah nyah..........)Sadly, the notes lack proofs, although most results are at least stated. Hill is more interested in these notes in explaining why certain results in  vector space theory are important-that all important "context". While the notes aren't completely applied, he does tend to downplay very abstract or lengthy proofs and most of the computational tools-such as matrices and determinants-are emphasized in the early lectures. He does give careful definitions and many good examples, including unusual applications such as to network flow. There's also emphasis on the role of projections in Euclidean space and cofactors.Also, there's very nice, if computational exercises and practice exams.If you're looking for an abstract course in linear algebra with everything proven carefully-well, forget it, you're better off with Cameron, Kerr, Stoll or some of the others here. But if you're looking for a good pedestrian overview of the elements of the subject, it'll make a very good supplement to those notes. Recommended to beginners in linear algebra. .
  72. Linear Algebra I Birne Binegar University of Oklohoma Spring 1999
  73. Linear Algebra II Birne Binegar University of Oklohoma Spring 2013  (PG/PG-13) These are the course materials for a 2 semester comprehensive course on linear algebra for mathematics majors-which, similarly to Ulrich Meierfrankenfeld's lectures, are a number of years apart in composing.  It's interesting because despite being somewhat lower in level to Meierfrankenfeld's course, they are quite similar in content, style and coverage.  Contents: (Linear Algebra I) Lecture 1: Vectors and Vector Spaces Lecture 2: The Geometry of Vectors Lecture 3: Matrices and Matrix Algebra Lecture 4: Matrices and Matrix Algebra, Cont'd Lecture 5: Solving Linear Equations Lecture 6: Inverses of Matrices Lecture 7: Subspaces of Rn  Lecture 8: Review Session Lecture 9: Linear Independence and Dimension Lecture 10: The Rank of a Matrix Lecture 11: Linear Transformations and Matrices Lecture 12: Abstract Vector Spaces and Concrete Examples Lecture 13: Determinants Lecture 14: Review Session Lecture 15: Eigenvalues and Eigenvectors Lecture 16: Diagonalization  (Linear Algebra II) : Lecture 1: Introduction: Fields and Vector Spaces Lecture 2: Subspaces, Linear Dependence and Linear Independence Lecture 3: Dimension and Bases Lecture 4: Elementary Operations and Matrices Lecture 5: Finitely Generated Vector Spaces Lecture 6: Quotient Spaces Lecture 7: Systems of Linear Equations Lecture 8; Solving Linear Systems Lecture 9: Homogeneous Linear Systems Lecture 10: Hyperplanes Lecture 11: Linear Transformations Lecture 12: Linear Transformations and Matrices Lecture 13: Homomorphisms and Isomorphisms Lecture 14: Endomorphisms and Automorphisms Lecture 15: Determinants Lecture 16: The Theory of A Single Endomorphism (Introduction) Lecture 17: Invariant Subspaces Lecture 18: The Triangular Form and Cayley-Hamilton Theorems As expected, there's some overlap between the notes because certain topics, such as linear independence and spectral theory, need to be covered at both levels for completeness.While as one would also expect, the basic notes are less rigorous then the "advanced" notes and focus more on the calculational aspects of the subject. Binegar defines everything carefully in either case and doesn't shy away from proving results when he deems them important.What's interesting here is that he takes multiple perspectives on vectors: geometric, algebraic and physical-and switches between them as needed. This supplies a great deal of much needed intuition by constructions in the plane and space as well as physics, in addition to abstract vector spaces. He also doesn't hesitate to omit proofs of results he feels are straightforward enough for students to do, such as the equivalence theorem and the properties of inverse matrices.  There are many careful and illuminating examples. The second set of notes, although considerably more sophisticated, lose none of the readability and care of the author's style. There are far fewer examples in the advanced notes.It's understandable as virtually all of those potential examples were already used in the elementary course and would simply be repeated here. But the author also indicates it's intentional on page 29: The one habit I've been trying to wean you of is the an over-reliance upon concrete examples to develop your understanding. He goes on to explain that at this higher level, he wants students to a) develop their own examples and b ) learn to work purely with the definition axioms in understanding the material. I have a lot of trouble agreeing with this,although many mathematicians share this opinion. Examples are the bedrock of mathematics upon which axiomatic systems are built. It's hard to image that the stock of examples one learned in an earlier course will be burned into the student's memory and that representing or referring to a few when learning the abstractions won't make it so much clearer and easier. Which is why these notes really should be read together as a unit for full effectiveness.Don't get me wrong, the second set of notes isn't devoid of examples, just much fewer present.  I still think for a year long course, the 2 notes would be most effectively used in tandem. In either case, Binegar has written a very lucid,complete and readable account of linear algebra Together, these 2 notes will form the foundation for an outstanding year long course in linear algebra for mathematics majors. Highly recommended to both students and teachers of linear algebra at all levels.
  74. Applied Matrix Algebra and Numerical Linear Algebra Laurent Demanet Stanford University Fall 2008  (PG)The notes and course materials for an applied course in linear algebra for applied mathematics students and majors in other disciplines. As expected, they focus on computational methods and algorithms-especially numerical approximation and solution of differential equations- and avoid hard proofs from the theory of abstract vector spaces. That's not to say they're completely pragmatic and don't care about proofs-things are carefully defined, major theorems are stated and simple proofs are given as homework assignments. But the parts of the theory the course concerns itself with is more specialized then the usual linear algebra courses-it emphasizes the analytic aspects of linear operators that form the theory of linear approximation. The textbook for the course was the superb  Numerical Linear Algebra by L. N. Trefethen and D. Bau III and the notes and exercises don't stray far from this material.Demanet's notes are elementary and readable and don't go too deeply into what's really a very sophisticated area of applied mathematics. The exercises are quite good as well. Since this is a very active and important area of both mathematics and computer science, students should get to understand the basics of it. This is as good a source as any. Highly recommended.
  75. Linear Algebra I Keith Matthews University  of Brisbane 1991
  76. ELEMENTARY LINEAR ALGEBRA K. R. MATTHEWS UNIVERSITY OF QUEENSLAND  Corrected Versionh April 2013 (PG)This is yet another excellent online textbook in basic linear algebra, one that the author has been revising based on his first year linear algebra notes at Queensland which he taught for nearly 2 decades.Contents:  Chapter 1: Linear Equations Chapter 2: Matrices Chapter 3: Subspaces Chapter 4: Determinants  Chapter 5: Complex Numbers  Chapter 6: Eigenvalues and Eigenvectors  Chapter 7: Identifying Second Degree Equations  Chapter 8: Three-dimensional Geometry  Further Reading/Bibliography Index  Corrections If you're looking for anything groundbreaking or original in this textbook or the 1999 original lecture notes from which it was developed, forget it. Matthews definitely isn't looking to reinvent the wheel here. The title really does say exactly what the book is about-no more, no less.That being said, he's crafted a very good wheel indeed. Matthews' presentation of the essentials of vector spaces and linear maps is very careful and rigorous with many examples-both computational and theoretical- and a smooth and simple prose. The presentation is purely mathematical-there are almost no applications. But the proofs and examples of the standard results of linear systems of equations, matrices, linear maps, vector spaces and subspaces, determinants, inner products, eigenvalues and eigenspaces and much more-is all done very well and would make an ideal first course for self study or a course. There are tons of good problems, most very straightforward. Best of all-all the solutions can be found here.There are better books available online, but it would be a mistake to pass on  Matthews' text. Highly recommended!
  77. Applied Matrix Theory Jacob Bernstein Stanford University 2010    (PG) This is Jacob Berstien's version of the same course commented on here taught by Laurent Demanet.As we said earlier, this is a specialized linear algebra course focusing on the analytic aspects for numerical approximation theory and related computer programming methods.One should warn the prospective user that Bernstien has several versions of the notes at the site-one should make sure one gets the most recent version (2010) as the earlier ones are riddled with errors. Again, the emphasis here is on the aspects of the theory of vector spaces most important in approximation and solution
  78. methods for large or complicated matrices, which appear a great deal in real world applications of matrix theory. The notes are a bit sloppy, by  the author's own admission-but I found them quite readable and interesting neverthless. The sections on th QR algorithm and the multivariable least squares algorithm are particuarly nice.The exercises are good as well-varying in difficulty from easy to fairly challenging, but none too difficult. Still, since the notes are so sloppy, maybe beginners should beware and stick to course texts. Teachers may wish to correct them and use them. So caveat emptor, people. Contents: Matrices, vectors and their products (review) Matrices as linear transformations  Rank of a matrix, linear independence and the four fundamental subspaces of a matrix Orthogonality and isometries The QR decomposition Eigenvalues and the spectral decomposition of symmetric matrices The singular value decomposition and its applications The conditioning of a matrix Least squares problems Algorithms for solving systems of linear equations and least-squares problems Iterative methods for solving linear systems: the method of conjugate gradients
    Lecture Notes on Linear Algebra A Second Course Ron Umble Millersville University of Pennsylvania Fall 2012 (PG-13) This is a highly unusual set of lecture notes for a second course in linear algebra.Additional handwritten notes, exercises and other materials that accompanied these notes can be found here.Contents: Computing the Inverse of an Invertible Matrix 5 Simpson's Rule 7 Vector Spaces and Linear Independence 9 Matrix Representation of Linear Maps 11 Geometry of Linear Operators on R2 13 Change of Coordinates in R2 17 Re‡flections in R2 and Rotations in R3 21 Inner Product Spaces and Orthogonal Transformations 23 The Fibonacci Sequence and Golden Ratio: An Application of Diagonalization 27 Orthogonal Diagonalization 31 Spectral Decomposition and Quadratic Forms 33 Space-Time: A Euclidean Pseudo-Inner Product Space 39 The Dimension Theorem 49 Similarity Invariants 51 Row-reduction to Hessenberg Form 53 Krylov’s Method –Characteristic Polynomials of Unreduced Hessenberg Matrices 55 Characteristic Polynomials of  Reduced Hessenberg Matrices 59
  79. Householder’s Method –Hessenberg Form via Orthogonal Transformations 63 Two Applications of Householder Transformations 7 Matrix Polynomials and the Cayley-Hamilton Theorem 79 Generalized Eigenvectors and Systems of Linear Differential Equations 83 Schur’s Triangularization Theorem 89 Another Proof of the Cayley-Hamilton Theorem 93 The Range-Nullspace Decomposition of Fn 95 Nilpotent-Nonsingular Form of a Singular Matrix 99 Jordan Form of a Nilpotent Matrix 103 Jordan Form of a General Matrix 113 What's unusual about these notes is that they don't fit any of the usual patterns for a second course in linear algebra. They clearly aren't a typical "advanced linear algebra" course i.e. a completely rigorous and abstract treatment of basic linear algebra from the standpoint of modern mathematics, although there is a great deal here that should be present in such a course. They aren't  a "matrix analysis" course focusing on matrix approximation theory like Demanet or Bernstien's, although some of that material is present here. They aren't an "applied linear algebra" course at the graduate level, either. So what are they about? What Umble's amazingly written here as a strange hybrid of all three kinds of courses-the applications are completely intertwined with a very advanced, sophisticated theoretical presentation of linear algebra from an abstract point of view.I was incredibly excited when I realized what he was intending on writing here-although I was wondering if he could pull it off.He indeed does-the resulting notes are strikingly original, readable and contain a wealth of material that one doesn't find contained in standard linear algebra texts even at the advanced level. There are not only many wonderful computational examples and careful proofs of theorems, there are a ton of terrific applications to physics and geometry, applications one doesn't see except in very specialized applied courses.The table of contents gives some idea of the wonderfully rich diet these notes will provide a student for thier second course in linear algebra. The amazing part is that even with this range of coverage, Umble manages to bring the notes in at a paltry 122 pages! His style is careful and concise, but pleasant and very friendly. This is one of the most complete and original sets of lecture notes I've ever seen and combined with one of the more standard advanced linear algebra courses listed here, it will serve as the foundation of a superior advanced vector space course. Don't pass this by. The highest possible recommendation for students and teachers of linear algebra!
  80. Advanced Linear Algebra Kevin Purbhoo University of Waterloo Spring 2012  (PG-13) Another set of notes for a second course in linear algebra from an advanced point of view. Contents: Matrix algebra: determinants, eigenvectors, companion matrices, exponential of a matrix, stochastic matrices; Vector space constructions: direct sums, dual spaces, quotient spaces, free vector spaces; Canonical forms: Cayley-Hamilton theorem, minimal polynomial, primary decomposition theorem, Jordan canonical form, rational canonical form; Inner product spaces: projections, Gram-Schmidt orthogonalization process, pseudo-inverse, isometries, normal operators, spectral theorem; Bilinear and multilinear algebra: tensor products, the exterior algebra, quadratic forms, symplectic forms. These are more traditional and focus on the "hard algebraic" aspects of more advanced vector space material. They are readable, concise and clear with a number of examples. There are also aspects that are unusual and contain some quite important topics, such as the matrix exponential, a brief but very readable treatment of tensor products and multilinear algebra and symplectic forms. The main flaw in these notes, besides several minor errors, is that exercises sets are missing. Again, this is a big problem for self study. Still, I think they're quite nice and will serve either an honors student or a first year graduate student quite well in learning the more advanced topics of linear algebra quickly. Solid but not spectacular. Recommended. Advanced Linear Algebra Bruce E. Shapiro California State University Fall 2012 (PG-13) This is another one of Shapiro's excellent, substantial lecture notes, this one for a second course in linear algebra assuming the students know the basics of matrix algebra, vector spaces and linear maps.Contents Front Cover Table of Contents Symbols Used 1 Complex Numbers 2 Vectors in 3-Space 3 Matrices and Determinants 4 Eigenstuff 5 Inner Products and Norms 6 Similar Matrices 7 Previewing the SVD 8 Example: Metabolic Flux 9 Vector Spaces 10 Subspaces 11 Polynomials  12 Span and Linear Independence 13 Bases and Dimension 14 Linear Maps 15 Matrices of Linear Maps 16 Invertibility of Linear Maps 17 Operators and Eigenvalues 18 Matrices of Operators 19 The Canonical Diagonal Form 20 Invariant Subspaces 21 Inner Products and Norms 22 Fixed Points of Operators 23 Orthogonal Bases 24 Fourier Series 25 Triangular Decomposition 26 The Adjoint Map 27 The Spectral Theorem 28 Normal Operators 29 Positive Operators 30 Isometries 31 Singular Value Decomposition 32 Generalized Eigenvectors 33 The Characteristic Polynomial 34 The Jordan Form Like his other note sets, it's very comprehensive, rigorous and innovative in it's choice of material, while simultaneously a joy to read. It's extremely readable and friendly-and actually quite amusing in places. For example, note the cartoon on Bruce's Law in the preface section-which should make most of us who have survived the academic wars chuckle.  He's not afraid to use imprecise intutive language as long as it's made precise later on in the text or to use terms from basic linear algebra without definition. For example, he defines vectors in R3  as ordered triples with magnitude and direction-and then immediately assumes the students knows what the inner product and transpose of matrices are.It's clear when the author defines these terms, it's strictly for review and/or to set the notation-the student is assumed to be familiar with them.Complex vector spaces are used throughout, which enhances the presentation by giving important specific topics such as
  81. Hermitian matrices.Many minor but useful topics are present that usually aren't in either basic or advanced linear algebra courses,such as a full proof of Cramer's rule and induced matrix norms with geometric interpretations in the plane. There are also a number of unusual applications, such as singular matrix decomposition to model the kinetics of metabolic flux-which as a former biochemistry major, I found fascinating. There's a ton of detailed examples, especially in difficult topic sections. My one complaint-once again-no exercises. That's really what demonstrates these notes aren't really ready for prime time as a stand alone text-and Shapiro admits as much in the introduction. Theofficial text for the course was Sheldon Axler's Linear Algebra Done Rightwhich I thought was kind of an odd choice given Axler's very careful but unorthodox presentation and lack of applications. Personally, I would have chosen either the classic Linear Algebra An Introductory Approach by Charles Curtis or the equally good and more comprehensive Linear Algebra by  Stephen H. Friedberg ,Arnold J. Insel  and Lawrence E. Spence. Either would have complemented Shapiro's approach better due to the less abstract presentation and the presence of many important applications, especially in Friedberg, et. al. In any event, Shapiro has written another winner here for an advanced linear algebra course and I really hope he continues to polish it so that one day it can be published as a for-real textbook and hopefully he'll continue to make freely available online after that. Very highly recommended for advanced students and teachers.
  82.  Linear Algebra II  Yaroslav Vorobets Texas A & M University Spring 2012 (PG-13) Very comprehensive PowerPoint style PDF notes for a second course in linear algebra based on Linear Algebra by  Stephen  H. Friedberg ,Arnold J. Insel  and Lawrence E. Spence. Like that excellent text, they begin at the beginning and therefore can also be used for an honors course. Contents: • Vector spaces • Linear transformations • Systems of linear equations, matrix algebra, determinants •Eigenvalues and eigenvectors, diagonalization • Inner product spaces, special classes of operators • Jordan canonical form  The level of detail is impressive and when combined, they are over 600 pages in length-which means they effectively form their own textbook without exercises!  However, they follow the text very closely and don't really strike any original notes. That being said, they are very carefully written and readable, have lots of diagrams and help to clarify some parts of the text that aren't so good, such as canonical forms and the Jordan form-which sadly, I think is a mess in an otherwise outstanding textbook. Vorobets' presentation of this very difficult but critical topic is somewhat better then the text, but not much. The notes also have many excellent examples that are not in the text that really show how the axiomatic approach to linear algebra can give some very counterinitutive results-i.e. the example of subspaces of infinite sequences in R?  These notes will serve a student taking an honors linear algebra or second course in self study quite well, but I'd recommend supplementing them for some sections with one of the other note sets here, such as Umble, Cameron or Treil. And of course, problems will be needed. I think pairing these notes with Erdman's problem course would make an outstanding advanced text in linear algebra for self study. A good, versatile set of notes for students in linear algebra. Highly
  83. recommended.
  84. Linear Algebra I J. Lawlor University of Vermont Spring 2013 (PG) Extensive set of handwritten lecture notes for a first semester undergraduate course.While the coverage is broader then usual and the notes are nice and readable with many solved examples, the choice of topics is very standard and doesn't really present anything you can't find in Hefferon, Beezer, Kerr or a half a dozen other online texts or lecture notes available here-and frequently is presented better in those sources.Feel free to browse them-but really nothing special here.
  85. Linear Algebra T. Kapitula  University of New Mexico  2004 (PG) Another very standard set of notes for a first course in linear algebra. In fact, these notes are designed to supplement Jim Hefferon's textbook. Which begs the question-why write an online supplement for a textbook that's available online for free? Probably to provide an alternate perspective from the teacher and to clarify certain sections. Which is fine by me. The supplement is well written, nice pictures and an ample supply of examples. But to be honest, I don't think they add much to Heffron's excellent text. Free free to browse and use them-but if you don't and just use Hefferon's book or one of the other recommended sources here- I don't think you'll be worse off for it.
  86. Introduction to Linear Algebra T. Scofield Calvin College April 11, 2013 (PG)Yet another set of lecture notes for a standard first course in linear algebra.However, the organization of these notes is a bit unorthodox and therefore worth scrutiny.Contents 1 Solving Linear Systems of Equations 1.1 Matrix Algebra 1.2 Matrix Multiplication and Systems of Linear Equations 1.2.1 Several interpretations of matrix multiplication 1.2.2 Systems of linear equations 1.3 Affine transformations of R2 1.4 Gaussian Elimination 1.4.1 Examples of the method 1.4.2 Finding an inverse matrix 1.5 LU Factorization of a Matrix 1.6 Determinants 1.6.1 The planar case 1.6.2 Calculating determinants for n-square matrices, with n > 2 1.6.3 Some facts about determinants 1.6.4  Cramer’s Rule 1.7 Linear Independence and Matrix Rank 1.8 Eigenvalues and Eigenvectors 2 Vector Spaces 2.1 Properties and Examples of Vector Spaces 2.1.1 Properties of Rn 2.1.2 Some non-examples 2.2 Vector Subspaces 2.3 Bases and Dimension 3 Orthogonality and Least-Squares Solutions 3.1 Inner Products, Norms, and Orthogonality 3.1.1 Inner products 3.1.2 Orthogonality 3.1.3 Inner product spaces 3.2 The Fundamental Subspaces 3.2.1 Direct Sums 3.2.2 Fundamental subspaces, the normal equations, and least-squares solutions 4 Selected Answers to Exercises 105 They are clearly not intended as a rigorous course-all but the simplest theorems are stated without proof.  The study of eigenvalues and eigenspaces preceeds that of abstract vector spaces. This really points out how calculational aspects of the subject are front and center here. However, all definitions and theorems are carefully stated and there are an enormous number of detailed examples, some highly unusual.There's also a lot of pictures and most importantly, an unusually large number of exercises for students to chew on of varying levels of difficulty.Lastly, there are many applications to computer science, including inserted code for MATHLAB and Sage programs.  The lack of detailed proofs, however, can be a positive for teachers or active learners-it makes the notes quite versatile in how much rigor you really want to add yourself. Supplemented with a more theoretical treatment, Scofield's notes can indeed serve as the basis of an excellent first course in linear algebra for mathematics majors. Applied math majors could simply use the notes as is. In any event, a useful and interesting set of notes for a first course .Recommended.
  87. Linear Algebra II George Melvin University of California, Berkeley Summer 2012  (PG-13) These detailed,very careful and engrossing notes were written for a summer advanced course in linear algebra at Berkeley.Contents 0 Preliminaries 0.1 Basic Set Theory 0.2 Functions 1 Vector Spaces & Linear  Morphisms 1.1 Fields 1.2 Vector Spaces 1.2.1 Basic Definitions 1.2.2 Subspaces 1.3 Linear Dependence & span sms, Part I 1.5 Bases, Dimension 1.5.1 Finding a basis 1.6 Coordinates 1.6.1 Solving problems 1.6.2 Change of basis/change of coordinates 1.7 Linear morphisms II 1.7.1 Rank, classi cation of linear morphisms 1.8 Dual Spaces (non-examinable) 1.8.1 Coordinate-free systems of equations or Why row-reduction works 2 Jordan Canonical Form  2.1 Eigenthings 2.1.1 Characteristic polynomial, diagonalising matrices 2.2 Invariant subspaces 2.3 Nilpotent endomorphisms 2.3.1 Determining partitions associated to nilpotent endomorphisms 2.4 Algebra of polynomials 2.5 Canonical form of an endomorphism 2.5.1 The Jordan canonical form 3 Bilinear Forms & Euclidean/Hermitian Spaces 3.1 Bilinear forms 3.1.1 Nondegenerate bilinear forms 3.1.2 Adjoints 3.2 Real and complex symmetric bilinear forms 3.2.1 Computing the canonical form of a real nondegenerate symmetric bilinear form 3.3 Euclidean spaces 3.3.1 Orthogonal complements, bases and the Gram-Schmidt process 3.4 Hermitian spaces 3.5 The spectral theorem 3.5.1 Normal morphisms  3.5.2 Self-adjoint operators and the spectral theorem  The author is a graduate student who seems-from both this and his webpage-takes great delight in teaching and encouraging the dedicated mathematics majors there. (They have to be dedicated-otherwise, why would they be taking such  a challenging course in the sweltering summer heat?!?)The notes have a wonderful personal, chatty style one rarely sees in books or notes at this level-they're an absolute joy to read. I really shouldn't, but to give an idea of the joy of the author's prose, I'll quote you one small example from early in the course:   Example 0.2.3. Consider the set P described above (so an object in P is a person in Etcheverry,room3109, at 10.10am on 6/18/2012) and let C denote the set of all possible cookie ice cream sandwiches available at C.R.E.A.M. on Telegraph Avenue (for example, vanilla ice cream on white chocolate chipcookies). Consider the following functionf : P ? C ; x ? f (x) = x's favourite cookie ice cream sandwich.In order for f to de ne a function we are assuming that nobody who is an element of P is indecisive sothat they have precisely one favourite cookie ice cream sandwich.So, for example,f (George) = banana walnut ice cream on chocolate chip cookies.  I bet most of us wish we'd had functions explained to us that way our first time out! The author has a real talent for entertaining and  engaging his audience, but none of that means anything if he can't instruct them in the process. I'm pleased to report he does more then that. He presents the material in not only a rigorous,relatively detailed and lucid manner, but provides many beautiful insights and sidebars designed to get beginners used to dealing with rigorous mathematics.There are many wonderful and nonstandard examples,such as  the trivial vector space on a finite set and the set of all 3 x 3 matrices of trace 0,which Melvin doesn't hesistate to connect to other, deeper areas of mathematics or historical notes about thier origin.There  also many terrific exercises of varying type and difficulty at Melvin's homepage for the course. There are no applications, these notes are pure algebra-but that's fine for a course at this level and you can always supplement it with applications. Indeed, Umble's or Berstien's lectures provide sophisticated applications appropriate for this level course.I think these notes have very rapidly become my favorite online source for a second course in linear algebra. Melvin's notes do what the very best mathematical exposition does-they not only inform, but inspire the neophyte and make coming to class a joy. If only all of us had that ability. At the very least, we have teachers like Melvin to show us how. The highest possible recommendation!
  88. THE LINEAR ALGEBRA PRIMER MULTILINEAR REGRESSION Maurice J. Dupre Tulane University April 2009 (PG) Very brief (9 pages) overview of linear algebra and multilinear maps in preparation for an advanced course in probability and statistics. Despite thier brevity, the author does a very nice job reviewing the sophisticated definitions and main results of linear algebra. Useful for review. 
  89. Numerical Linear Algebra Rudi Weikard University of Alabama at Birmingham 2010 (PG-13) Careful and precise, but quite dense and difficult  notes for what's usually presented as a strictly non-rigorous course for advanced students of applied mathematics. Weikard begins with an intensive review of the aspects of the abstract theory of linear spaces of greatest importance in the theory of matrix approximation, such as the operator norm and the singular value decomposition of matrices. From there, he tackles most the major basic methods of numerical linear algebra, always stating everything carefully and showing how proofs ultimately are the source of all the algorithms. Weikard seems to believe what I've always thought-the best practitioners of applied mathematics always have a very firm grasp of theory and this fact should be reflected far more often in the teaching of graduate students then it usually is. The notes only cover the basics, but what they do cover, they cover better then most online sources do, with a great deal of care and insight. Highly recommended for graduate students. Chapter 1. Numerical Linear Algebra 1.1. Fundamentals 1.2. Error Analysis 1.3. QR Factorization 1.4. LU Factorization 1.5. Least Squares Problems 1.6. Eigenvalues 1.7. Iterative Methods 1.8. Problems 1.9. Programming Assignments Index
  90. SECOND YEAR LINEAR ALGEBRA JESSE RATZKIN University of Cape Town 2010 (PG) Notes for a beginning course in linear algebra that assumes only the  barest elements of matrix manipulation and emphasizes the role of linear transformations in Euclidean geometry, which is an effective way to teach the subject. By the author's own admission in the preface, he's "shamelessly cribbed these notes" from Linear Algebra by  Stephen H. Friedberg ,Arnold J. Insel  and Lawrence E. Spence. Being very familar with this book, I can testify he's not exaggerating- more or less the entire content of the notes appears nearly word for word at some point in Friedberg, et. al.  Which isn't necessarily a bad thing as that book-as outstanding as it is-is quite lengthy and to use it as a course text, an instructor needs to be selective with it. Ratzkin has done the selecting here and he's done a good job-the prose is sharp and clean, there are lots of pictures in the plane and the focus is clearly on Euclidean geometry via isometries and similarities. Inner product spaces and their finite dimensional geometry are developed in detail. No,it's not original-but the author succeeds in extracting and organizing the desired topics from the text and if you're pressed for time and need a good, relatively short source on linear algebra in geometry, this isn't a bad choice. It won't win any awards, but geometry and linear algebra students will certainly find it convenient and helpful as a focused study aid. Recommended.
  91. Numerical Linear Algebra Nikolai Chernov University of Alabama at Birmingham  2014 (PG-13) An excellent, very lucid and detailed presentation of numerical linear algebra for students who have had at least a basic undergraduate course in linear algebra.Contents 0. Review of Linear Algebra 1. Norms and Inner Products 2. Unitary Matrices 3. Hermitian Matrices 4. Positive Definite Matrices 5. Singular Value Decomposition 6. Schur Decomposition 7. LU Decomposition 8. Cholesky Factorization 9. QR Decomposition 10. Least Squares 11. Machine Arithmetic 12. Condition Numbers 13. Numerical Stability 14. Numerically Stable Least Squares 15. Computation of Eigenvalues: Theory Most presentations of approximation methods by matrices and linear operators are extremely dense and specialized. That's a shame because this is an incredibly important area of both mathematics and computer science for the analysis and solutions of problems in the physical sciences and engineering. Chernov's notes are a very welcome change as they are not only very careful, they are comprehensive, superbly organized and best of all, readable. Very few sources on this topic succeed in being all of these things at once. For example. Wiekard's notes are quite careful, but nowhere near as accessible or broad in scope as Chernov's. They also assume quite a bit more background then Chernov's., whereareas these notes carefully develop all the advanced linear algebra needed and only require a basic one semester course as background. There are also many important topics here that are usually presented in much more advanced treatises, such as machine arithmetic and eigenspace stability on manifolds. These are by far my favorite source on this important topic.I highly recommend it to anyone interested.
  92. Orthogonal Bases and the QR Algorithm Peter J. Olver University of Minnesota (PG-13)A relatively brief but clear and careful account of the QR algorithm and it's foundations in orthogonal inner product spaces. Has Oliver's usual computational yet careful style, which will be very helpful for students at all levels who have had good courses in linear algebra and nonrigorous calculus. Highly recommended as either a supplement to a linear algebra course or an independent study project.
  93. Linear Algebra Matrices Christopher Cooper McQuarrie University (LAM)(G)
  94. Vector Spaces Christopher Cooper McQuarrie University (VC)(PG) These are outstanding notes for a 2 semester course on linear algebra which is developed in a very innovative way. The first half of the course (LAM) focuses on vectors and matrices in Rand R3 in a unified manner-vectors in the plane and space are column vectors of 2 x 2 and 3 x 3 real entry  matrices respectively-which he then generalizes to m x n matrices. In these notes, Cooper builds and proves all aspects of linear algebra that can be developed purely in terms the vector space of of m x n real valued matrices-the space of which,of course, is isomorphic to Rmn -without abstract vector space machinery. This turns out to be a surprisingly large amount.  Cooper gives a very large number of examples focused in the plane and space, including many detailed computations. Also, the first half is surprisingly careful, as he proofs all results that can be states and proved in this concrete setting that generalize without change to abstract spaces. Indeed, there is a surprisingly careful presentation of eigenspaces and diagonalisation completely in concrete spaces that clarifies enormously the geometric aspects of these methods of solution. He also gives a number of creative and diverse exercises that either prove minor results, do computations or give applications of the material-such as solving the characteristic equation of the stochastic matrix that predicts the probable path of a robot vacuum cleaner. The second course notes (VC) resemble quite a bit more a conventional rigorous course in linear algebra, but it builds on the first set of notes and assumes as known all the material covered in there.This allows Cooper to move quickly to recast all the concepts of LAM in terms of abstract vector spaces, their subspaces and the linear maps between them. Included here are such uncommon fare for a second course as direct sums of vector spaces and entire chapters on abstract quadratic forms and numerical linear algebra. The last can serve as an excellent brief introduction to more comprehensive treatments such as Chernov or Wiekard. In each case, Cooper's style is concise yet wonderfully lively and insightful-his presentation has many pictures and subtle points that are usually omitted or downplayed. There are lots of examples and pictures in both notes. They are substantial but focused-they don't try and do everything.The concrete to abstract transition between the notes will be very helpful, especially for students that use both in tandem. Best of all, in each set of notes, there are complete solutions to each chapters' exercises. I can't think of a better introduction to the subject for the serious self studying mathematics undergraduate.  Most highly recommended, particularly for self study! There are sources here that can match it,but none that can surpass it in both quality and clarity.   Contents: (MC) Introduction and Contents CHAP01 Vectors in 2 Dimensions CHAP02 Matrices in 2 Dimensions CHAP03 Vectors and Matrices in 3 Dimensions CHAP04 Matrices and Determinants CHAP05 Systems of Linear Equations CHAP06 Eigenvalues and Eigenvectors CHAP07 Diagonalisation  (VC)  Introduction and Contents  CHAP01 Vector Spaces   CHAP02 Linear Transformations   CHAP03 Inner Product Spaces  CHAP04 Diagonalisation Revisited  CHAP05 Quadratic Forms CHAP06 Numerical Linear Algebra  CHAP07 Jordan Canonical Form
  95.  Linear Algebra II Michael Bate University of York 2011-2012 (PG) Another abstract and careful first/second course in linear algebra in the UK style. It can be used for either an honors course or a second course in linear algebra in the US for students with some mechanical exposure to matrix algebra. Relatively brief and terse, but contains careful proofs and some nicely selected examples.Contents  Vector Spaces Bases and Dimension Subspaces Linear Transformations Matrices Coordinates and Matrices No applications, strictly for pure mathematics students.There are some original touches-tensor products are introduced very early. The main drawback of these notes is that there are no exercises. None. Given the fact these notes don't really present anything strikingly original or deep, the lack of exercises really hurts their value. I'd use Stoll or one of the other advanced note sets.
  96. Linear Algebra I Ronald van Luijk University of Liedien 2013  (PG) Yet another abstract first course in linear algebra from outside the US. If the notes seem very familiar when reading them, they should-by the author's own admission, the bulk of these notes were copied from Micheal Stoll's extensive notes for the same course at Liedien. There's really not much added to them, so you may as well download the excellent originals by Stoll, which cover a great deal more and are even more readable because they're complete. Take a look if you want, but you've been warned.
  97. Linear Algebra II Peter M Neumann Queen’s College University of Oxford (PG-13) A substantial,rigorous second course in linear algebra,again in the concise "Oxford" UK style.Again, this is a purely mathematical treatment of the subject-those looking for applications need to look elsewhere.While the notes are concise, they cover a surprisingly large amount of ground, including very detailed discussion of the motivation for the Cayley-Hamilton theorem,the theory of inner product spaces and a derivation of Bessel's inequality using Gram-Schmidt orthogonalization. Neumann writes very well indeed-the notes are not only very lucid and enlightening as well as careful. he injects interesting historical notes on the founders of the subject-often with his own unique dry British wit.As an example, consider his following reflection on Arthur Cayley's thinking that he didn't need a proof of the above famous result:  What a charming piece of 19th Century chutzpah: “I have not thought it necessary to undertake the labour of a formal proof of the theorem in the general case of a matrix of any matrices, and sketches for the 3×3 case could be practical for the n×n case: even if one could write down explicitly the characteristic polynomial of an n×n matrix A, it seems unrealistic to expect to write down the (i, j) coefficient of a general power Ak for all k up to n, and then evaluate cA(A) in the way that it is possible in the 2×2 case. So the fact is, he can’t really have had a proof. But there’s another fact: he may not have appreciated rigorous thinking in the same way as Oxford students now do (the poor chap went to Cambridge and missed out on the Oxford experience) but he did have a wonderful insight. He knew that the theorem was right, and for him proof, though it would have been nice to have,was less important than having an insight which he could use in all sorts of ways to solve other mathematical problems. 
    Delightfully entertaining sidebars like this are present all through the notes.There are also many wonderful exercises interspersed throughout the presentation, which vastly expand the range of the notes by making the students prove corollaries to the main results and give many additional computations and examples. While these aren't the most comprehensive of sources on rigorous linear algebra, what they do cover, they cover marvelously. I'm not sure if I could assign them as the only source for a course or  self study on linear algebra, but I'd certainly recommend them as a supplementary source. Highly recommended! CONTENTS Part I: Fields and Vector Spaces Fields Vector spaces Subspaces Quotient spaces Dimension (Revision) Linear transformations (Revision) Direct sums and projection operators Linear functionals and dual spaces Dual transformations Further exercises Part II: Some theory of a single linear transformation on a finite-dimensional vector space Determinants and traces The
  98. characteristic polynomial and the minimal polynomial The Primary Decomposition Theorem Triangular form The Cayley–Hamilton Theorem Further exercises Part III: Inner Product Spaces Real inner product spaces and their geometry Complex inner product spaces The Gram–Schmidt process Bessel’s Inequality The Cauchy–Schwarz Inequality Isometries of inner product spaces Representation of linear functionals Further exercises III Part IV: Adjoints of linear transformations on finite-dimensional inner product spaces Adjoints of linear transformations Self-adjoint linear transformations Eigenvalues and eigenvectors of self-adjoint linear transformations Diagonalizability and the spectral theorem for self-adjoint linear transformations An application: quadratic forms Further exercises
  99. Geometry and Linear Algebra Martin Liebeck  Imperial College London 2005   (PG) A very nice handwritten set of lectures and exercises that cover a unified first course in classical geometries and linear algebra. Indeed, I was conflicted initially about where to place them. I ended up listing it here under Linear Algebra, but the geometry part of the course suggests it could have just as easily been filed under Euclidean and Non-Euclidean Geometry. Matrix form transformations and calculations are used throughout to describe the geometry of the plane once the basics of
  100. linear algebra-linear systems, matrices, determinants, vector spaces, linear transformations-are established. Good exercise sets and exams. A very good course for the beginner giving a linear algebra course motivated by the intuition of classical geometry. Highly recommended.
  101. Linear Algebra II Simon Salamon Oxford University 2000  (PG)Yup, you guessed it-yet another set of lecture notes for the abstract second linear algebra course at Oxford.The coverage of topics is very similar (again, no pun intended) to Neumann's or Lauder's versions of the same course. However, Salamon's version is far terser and drier then either of those notes and his proofs of major results are sometimes confusing. For example, I've read his proof of the determinant product theorem twice and I still don't get his reasoning completely. Some sections are good and informative-such as his presentation of diagonalizability and it's relation to Euclidean geometric transformations. There's also a lot of good problems in the exercise sheets. Still, I think this material is covered much better in Neumann's or Cameron's notes, so I'd prefer those instead. Not bad,but not exceptional,either.
  102. Linear Algebra Mark Reeder Boston University Fall 2011  (PG) Concise,but comprehensive and mostly standard set of lecture notes and exercises-actually an online textbook by Reeder in progress-in basic linear algebra. Contents Syllabus Chapter 1  Arithmetic of 2x2 Matrices  Chapter 2  Special Types of Matrices  Chapter 3  Determinant and Trace  Chapter 4 Matrices as Linear Maps Chapter 5  Reflection Matrices Chapter 6 Fibonacci Numbers   Chapter 7  Migration Chapter 8  Eigenvalues and Eigenvectors  Chapter 9 Multiple Eigenvalues and Nilpotent  Matrices Exam 1 study problems  solutions Exam 1 solutions Chapter 10  Complex Eigenvalues Chapter 11, with solutions The geometry of the determinant and the Iwasawa decomposition  Chapter 12  Differential Equations Chapter 13  Vectors in Three-Dimensional Space Chapter 14   Three-by-Three Matrices and Determinants Chapter 15 The Kernel of a Three-by-Three Matrix Chapter 16  Three-by-Three Eigenvalues and Eigenvectors Chapter 17  Orthogonal Matrices and Symmetries of Space Chapter 18  Introduction to Four Dimensions  Exam 2 study problems with solutions Exam 2 Chapter 19  General Matrices    Chapter 20 Vector Spaces and Bases Chapter 21  The dimension of a vector space Chapter 22 Subspaces, Linear Maps and the Kernel-Image theorem  Chapter 23  Change of Basis  Final Study Problems They're very solidly written with many examples and good exercises of both a computational and theoretical nature. Also, like Cooper and Lerner's "texts", they take a concrete-to-abstract organization approach- Reeder doesn't develop abstract vector space theory until the end of the lectures and first develops the bulk of the concepts in low dimensions. The emphasis in the notes is on the role of linear transformations in classical geometry i.e. rotation, reflection and magnification matrices,which of course is a very natural path to take when you develop the material in detail in R2 and R3. He gives a very good introduction to the symmetry groups in the plane and space, a very important topic that's usually reserved for either abstract algebra course or specialized geometry courses. There are also several unusual applications, such as modeling migration patterns of populations using eigenspaces of systems of linear differential equations and the Iwasawa
  103. decomposition of an invertible matrix and it's role in determining orientation. (The latter application is particularly important in both geometry and algebra and  I wonder why most introductions avoid it-it's not that difficult! ) I do wish, especially given the emphasis on geometry,  that the author had included more pictures in the plane or space.But the reason for him not doing so is probably because many of the exercises call for the student to draw a picture of the described linear mapping! Overall,though, this is a quality-if not spectacular-choice for an online textbook. Highly beginners.
  104. Linear Algebra - As an Introduction to Abstract Mathematics Free online text by Isaiah Lankham, Bruno Nachtergaele and Anne Schilling University of California at Davis (PG/PG-13) Another online textbook in basic linear algebra that rigidly separates out the abstract vector space theory and the computational aspects of matrices and determinants. Anyone else beginning to see a pattern here? Unfortunately, it's a cold reality of the deteriorating higher education system of the US.Universities are concentrating less on teaching students actual math and  more on how many bodies they can cram into courses by offering the most watered down, pragmatic courses they can. The authors here have attempted to strike a bargain between Mammon and the Muse by this rigid separation that will allow the text to be used for either a pragmatic plug and chug applied course or a purely rigorous abstract course or a course that falls somewhere in between. The enormous versatility of possible courses gained by the format is a very strong argument for it's existence and would be even if the intentional dumbing down of linear algebra in this country wasn't occurring. This is why I suspect the majority of texts on linear algebra-both online and in print-will follow this format in coming years. The chosen material for either half is for the most part standard and presented clearly if routinely, with lots of good examples, pictures and exercises. Some unorthodox touches: A complete introduction to complex numbers and elementary complex functions as linear mappings in the plane, an unusually full short treatment of the elements of abstract algebra in appendices B1-B3 presumably for honors or advanced courses and the lengthy chapter 12 on matrix arithmetic and algebra, which is largely independent of the first 11 chapters and can be used for a short applied course on matrices for a diverse collection of students, from honors high school to graduate students in other disciplines. What really strikes you about this book is how versatile it is-a dozen or so very different kinds of possible courses can be constructed using it. As I've said-there's nothing original here, but it's well done and will serve both students and teachers well as a text or supplement in linear
  105. algebra. Highly recommended. Contents 1 What is Linear Algebra? Exercises 2 Introduction to Complex Numbers Exercises ii 3 The Fundamental Theorem of Algebra and Factoring Polynomials Exercises 4 Vector Spaces Exercises 5 Span and Bases Exercises 6 Linear Maps Exercises 7 Eigenvalues and Eigenvectors Exercises 8 Permutations and the Determinant of a Square Matrix Exercises 9 Inner Product Spaces Exercises 10 Change of Bases Exercises 11 The Spectral Theorem for Normal Linear Maps Exercises Supplementary Notes on Matrices and Linear Systems Exercises List of Appendices A The Language of Sets and Functions A.1 Sets A.2 Subset, union, intersection, and Cartesian product A.3 Relations A.4 Functions  Summary of Algebraic Structures Encountered B.1 Binary operations and scaling operations B.2 Groups, fields, and vector spaces B.3 Rings and algebras C Some Common
  106. Math Symbols & Abbreviations D Summary of Notation Used
  107. Linear Algebra A free linear algebra textbook and online resource written by David Cherney, Tom Denton and Andrew Waldron University of California At Davis (PG/PG-13)  ANOTHER free online linear algebra textbook?!? Yes, they're going to be ubiquitous in the coming years so get used to it-and for students and teachers of mathematics, having such a large variety of inexpensive texts to choose from for a required course is a very good thing! Very interestingly, this book as well grew out of courses given at UCD.There must be an educational initiative there that encourages professors to write online textbooks for their courses-which I'm all in favor of, if so.Since it's written for the same kind of course as Lankham,Nachtergaele and Schilling, I would naturally expect a considerable amount of overlap in content if not style.While there is some inevitable overlap, the style and scope of this course is very different.Contents 1 What is Linear Algebra? 2 Systems of Linear Equations 3 The Simplex Method 4 Vectors in Space, n-Vectors 5 Vector Spaces 6 Linear Transformations 7 Matrices 8 Determinants 9 Subspaces and Spanning Sets 10 Linear Independence 11 Basis and Dimension 12 Eigenvalues and Eigenvectors 13 Diagonalization 14 Orthonormal Bases and Complements 15 Diagonalizing Symmetric Matrices 16 Kernel, Range, Nullity, Rank 17 Least squares and Singular Values A List of Symbols B Fields C Online Resources D Sample First Midterm E Sample Second Midterm F Sample Final Exam G Movie Scripts Index Unlike the aforementioned previous version of the notes for this course, not only are abstract vector space methods blended with computational examples and techniques, the authors derive the computational techniques from the theoretical framework.The preface for these notes, in which the authors lay out thier philosophy and modus operandi for the book,is well worth reading by both students and teachers. What Cherney, et. al do in these notes they refuse to accept the idea that they need to separate out the mathematics from everything else in linear algebra-instead, what they do is work hard to make non-mathematics and beginning math student understand the reasoning behind the math and how the computational methods are a direct result of this abstract framework. The first 4 chapters develop the terminology and basic computational machinery of linear algebra: matrices and linear mappings and thier accompanying arithmetic and algebra. No real theory appears in these chapters, but it's really obvious it would be quite difficult to discuss the theory of vector spaces to beginners without this language. Chapter 4 gives a brief but detailed review of vector algebra in the plane and space in terms of matrices and the language of linear maps. Chapter 5 is really where abstract vector spaces first appear and the theory is developed from there in detail. Interestingly, they do assume the language of naive set theory is known to the student and they don't hesitate to use it.  For example, product spaces are defined via ordered pairs of vectors in chapter 5. There are an enormous number of computational and theoretical examples throughout the book with many innovative pictures-I don't think I've ever seen this many examples in a book before!!! This gigantic collection of special cases to the general definitions and results-a number quite unusual but simple-will be a huge asset to beginning students and teachers. There's also quite a few applications interspersed throughout which are given only after the theoretical basis is stated and proved-including the Simplex method of linear programming, matrix applications to computer graphics such as .gif files, binary code storage, volume computation of the parellelapiped and many, many more.  The exercises are intentionally difficult for the most part, but doable-most are written,according to the authors, for the students to solve together in groups. This creative touch will make the classroom an interactive environment, which sadly isn't done in mathematics as much as it should be. Another inventive touch is that a number of hyperlinks to internet sources of examples, side topics and applications are present in the book, which further broadens it's range. Most importantly and strikingly-the book is an absolute pleasure to read. The prose has personality as well as clarity and that makes learning a new subject so much easier. This online book is one of the very best currently available on any subject. I hope the authors continue to make it freely available and fervently recommend it for anyone about to take or teach a linear algebra course. The highest possible recommendation!