Linear Algebra Spence

[Dorian Aldrege]

Hoffman and Kunze is the classic rigorous linear algebra textbook. My old mentor Nick Metas was one of the team of graduate students at MIT who worked over the manuscript of the original lecture notes for the course. The book is brutally mathematical with very few examples.That being said,it's very carefully written with many details in the proofs and definitions. If you're willing to work hard and you're serious about learning linear algebra as pure mathematics,you can hardly do better. But you'll need to supplement it for exercises and examples.

Linear Algebra by Friedberg, Insel and Spence is probably the single most comprehensive linear algebra textbook on the market. It's extremely careful with a ton of examples and it blends pure theory with applications very well.It's far more detailed and readable then Hoffman and Kunze and contains many applications you won't find in other textbooks, such as stochastic matrices. It also has many wonderful exercises. I just have 2 minor quibbles with it. First,in some ways,it's too comprehensive-to use the book in a course,even a year long course,one would have to be quite selective with it. Second-the section on the Jordan form and the diagonalization procedure is simply put, a trainwreck. This is a really important topic,so this really hurts the book.

They use this fact to prove that the operator as no eigenvectors. But it seems to imply that any member of an infinite dimensional inner product space can be represented with a finite number of basis elements, since n and m are finite.

Edit: I should also mention that the notion of infinite sum in an inner product space doesn't make sense unless the space is also complete with respect to the induced norm, i.e. is a Hilbert space. In the nice situation a Hilbert space has an "orthonormal basis," which is not a basis in the linear algebra sense but in the sense that the span of the basis is dense.

Infinite bases have nothing to do with 'infinite linear combinations.' The reader who feels an irresistible urge to inject power series into this example [concerning vector spaces of polynomials] should study the example carefully again. If that does not effect a cure, he should consider restricting his attention to finite-dimensional spaces from now on. [!]

That exactly describes your situation: you have a vector space of polynomials in e^(it), and you're trying to insert power series. If you go back to the definition of "vector space," you should verify that the definitions and axioms allow you to form arbitrary finite linear combinations of vectors, but not "infinite" ones. Note that in certain vector spaces, wisely constructed infinite sums are okay. Those include Banach spaces and Hilbert spaces. However, your vector space is not a Banach space or a Hilbert space, so there's no concept of "infinite sum." (You would probably learn about Hilbert spaces and Banach spaces while studying real analysis or functional analysis.)

A second course in linear algebra builds on the concepts learned in an introductory course and delves deeper into advanced topics such as eigenvalues, eigenvectors, and diagonalization. It also covers applications of linear algebra in fields such as physics, engineering, and computer science.

Linear algebra has a wide range of applications in fields such as computer graphics, data analysis, cryptography, and machine learning. It is also used in solving systems of linear equations, optimization problems, and differential equations.

Linear algebra can be challenging for some students, as it involves abstract concepts and requires a solid foundation in algebra and calculus. However, with practice and a good understanding of the basics, it can be a rewarding and useful subject to learn.

There are many resources available for learning linear algebra, including textbooks, online courses, and video lectures. Some popular textbooks include "Linear Algebra and Its Applications" by David C. Lay and "Introduction to Linear Algebra" by Gilbert Strang. Online resources such as Khan Academy and MIT OpenCourseWare also offer free lectures and practice problems for learning linear algebra.

Can anyone suggest a relatively gentle linear algebra text that integrates vector spaces and matrix algebra right from the start? I've found in the past that students react in very negative ways to the introduction of abstract vector spaces mid-way through a course. Sometimes it feels as though I've walked into class and said "Forget math. Let's learn ancient Greek instead." Sometimes the students realize that Greek is interesting too, but it can take a lot of convincing! Hence I would really like to let students know, right from the start, what they're getting themselves into.

Does anyone know of a text that might help me do this in a not-too-advanced manner? One possibility, I guess, is Linear Algebra Done Right by Axler, but are there others? Axler's book might be too advanced.

For teaching the type of course that Dan described, I'd like to recommend David Lay's "Linear algebra". It is very thoroughly thought out and well written, with uniform difficulty level, some applications, and several possible routes/courses that he explains in the instructor's edition. Vector spaces are introduced in Chapter 4, following the chapters on linear systems, matrices, and determinants. Due to built-in redundancy, you can get there earlier, but I don't see any advantage to that. The chapter on matrices has a couple of sections that "preview" abstract linear algebra by studying the subspaces of $\mathbbR^n$.

I rather like Linear Algebra Done Right, and depending on the type of students you are aiming the course for, I would recommend it over Hoffman and Kunze. Since you seemed worried that Axler might be too advanced, my feeling is that Hoffman and Kunze will definitely be (especially if these are students who have never been taught proof-based mathematics).

Hands-down, my favorite text is Hoffman and Kunze's Linear Algebra. Chapter 1 is a review of matrices. From then on, everything is integrated. The abstract definition of a vector space is introduced in chapter 2 with a review of field theory. Chapter 3 is all about abstract linear transformations as well as the representation of such transformations as matrices. I'm not going to recount all of the chapters for you, but it seems to be exactly what you want. It's also very flexible for teaching a course. It includes sections on modules and derives the determinant both classically and using the exterior algebra. Normed spaces and inner product spaces are introduced in the second half of the book, and do not depend on some of the more "algebraic" sections (like those mentioned above on modules, tensors, and the exterior algebra).

From what I've been told, H&K has been the standard linear algebra text for the past 30 or so years, although universities have been phasing it out in recent years in favor of more "colorful" books with more emphasis on applications.

Edit: One last thing. I have not heard great things about Axler. While the book achieves its goals of avoiding bases and matrices for almost the entire book, I have heard that students who have taken a course modeled on Axler have a very hard time computing determinants and don't gain a sufficient level of competence with explicit computations using bases, which are also important. Based on your question, it seems like Axler's approach would have exactly the same problems you currently have, but going in the "opposite direction", as it were.

There were times when I was rather fond of Strang's Linear Algebra and Its Applications. I haven't looked at it for a long time, but back then I found it very clear and appealing. Even if you don't follow the book chapter by chapter, it might still give you ideas.

There is no ideal text for a beginning one semester course as taughtin the US to first or second year college students. Older books like H&K treat only the abstract theory, in a fairly conceptual way and (if I recallcorrectly) with maps written on the right contrary to what students do incalculus. A later generation of books like the original Anton are alsopure math books but start by overemphasizing unrealistic manipulations withsmall matrices and vectors; then there is an abrupt shift to abstraction.Determinants are presented in a purely computational mode, as though theywere really used for this purpose; then eigenvalues occur very late and againin oversimplified small examples. Fortunately the newer texts tend to mixpure and applied throughout, but as a result they contain far too much materialfor a first course. And eigenvalue theory still gets introduced very late.Strang is attractive in many ways, but too loosely written down and not suitable for an inexperienced reader without a reliable guide at hand. Asidefrom Strang, the emphasis in most US textbooks remains placed on unrealisticinteger calculations with very small matrices rather than on the geometry ofsubspaces, etc. The pervasive role of geometric thinking in the subject ismostly downplayed in texts, as is the role of analysis. For self-study,something like Friedberg-Insel-Spence may be the best compromise choice.

My old mentor Nick Metas was part of the teams of graduate students who worked over the drafts of H&K when they were writing it for the linear algebra course at MIT in the 1960's. That being said,despite its' rigor and beauty, I think a "pure" linear algebra course is just as big a mistake as a pure theoretical calculus course no matter how good the students are. It's like teaching music students all about pentamer, note grammar and acoustics and never teaching them how to play a single note. I don't go for this whole pure/applied distinction, it's an idiotic consequence of this age of specialization. I love rigor,but applications should never be denied or ignored. That's why my overall favorite LA text is Friedberg, Insel and Spence-it's the only one I've seen that aims for and hits a terrific balance between algebraic theory and applications. I also love Curtis for similar reasons, but it's coverage isn't as broad. I love books that aim for that Grand Mean Balance-sadly, in America, there aren't anywhere near enough such texts.

c80f0f1006