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I'm looking for a good abstract linear algebra text (i.e., not matrix crunching) for students who have completed a Strang-level linear algebra course plus exposure to a proof writing (e.g., induction and contradiction are not foreign concepts).

The course is for students who want to revisit core/essential linear algebra concepts from an abstract viewpoint (e.g., for preparation for functional analysis or understanding theories in machine learning) but only so far as this abstract viewpoint supports better applications (I'm not going the Halmos or Lang route, for example).

The student mix will likely be statisticians, operations research, compsci and engineering, physics, and perhaps some math majors (applied, most likely).

I perused Lax's book and I generally like its contents, especially the inclusion of a dynamics chapter and lots of cool appendices that show how the abstract viewpoint is not just elegant, but very useful. However, it's also rather thin (which is actually nice!) but makes me wonder how idisyncratic Lax's choices were for the material in the book.

Has anyone used this book in a class -- if so, how did you and your students like it? Did most seem to think the price of the book was worth it and planned to keep it for reference?

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    $\begingroup$ Just giving another suggestion, Linear algebra done wrong by Treil. $\endgroup$
    – Panda
    Dec 8, 2017 at 20:49
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    $\begingroup$ If you haven't already, surely looking at Roman's Advanced Linear Algebra or Insel Spence and Friedberg's Linear Algebra might be good. I'm not sure I have a copy of Lax, so I have nothing to say about that. $\endgroup$
    – James S. Cook
    Dec 9, 2017 at 4:20
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    $\begingroup$ I'm curious what you define as thin. My experience of textbooks on the whole is thin=UK, fat=US. $\endgroup$
    – Jessica B
    Dec 9, 2017 at 8:04
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    $\begingroup$ I like Lax's book, and think it fits well with the OP's stated objectives, in particular because it demonstrates good taste with regards to the inclusion of a lot of more applied material typically, but unfortunately, omitted in such texts (e.g. duality in chapter 2!). The mathematical level (in particular, the style of the exposition) is higher than in other texts suggested in the comments, possibly too high for some students from statistics, engineering, etc., but it could work well for motivated students with a solid prior background in basic linear algebra. $\endgroup$
    – Dan Fox
    Dec 9, 2017 at 12:57
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    $\begingroup$ I had a similar course as the one you are describing of "Advanced Linear Algebra" (basically Functional Analysis disguised as Linear Algebra) we used Meyer's Matrix Analysis and Applied Linear Algebra. Albeit it starts with matrices and number-crunching it quickly jumps to more abstract linear algebra. The thing I liked the most is that it is really focused on applications and has a lot of examples / exercises / projects which apply the concepts. $\endgroup$
    – Rodrigo Zepeda
    Dec 14, 2017 at 18:46

1 Answer 1

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I'd like to thank all the commenters for their insights. In the end, I have decided to go with Lax's Linear Algebra and its Applications, primarily due to its insistence on abstract vector spaces (i.e., the field is not assumed to be complex or real numbers) and the focus on linear transformations as opposed to matrices, which will set up the students nicely for functional analysis.

Also, my concerns about his "applied" examples have been allayed, as I think the students will find them interesting and not the typical "plug and chug" applications they get elsewhere. The focus on analysis-related topics will also help tie in some aspects of what they learned in calculus (e.g., $e^A$ as a Taylor series...cool connection between linear algebra and calculus!).

I doubt I'll go through the whole book for a semester -- despite it's brevity, it's very, very dense on knowledge (so in terms of $/bit, it's quite well priced ;-). I think that covering Chapters 1-7 is a must as core abstract material.

  1. Fundamentals (basic definitions of abstract linear spaces and quotient spaces)
  2. Duality -- I'm including this because I want to hit on spaces of functions at the end wrt machine learning, where the idea of a dual space is important.
  3. Linear Mappings
  4. Matrices
  5. Determinants and Trace
  6. Spectral Theory
  7. Euclidean Structure (lots of important and very widely applied concepts here)

Also, given the interest in machine learning among students this group, I also think that Chapters 8,9,10, and 16 should be delved into, since symmetric and/or positive matrices play a role in many machine learning algorithms.

  1. Spectral Theory of Self-Adjoint Mappings: Covers the common case of symmetric matrices (e.g., in PCA and aspects of SVD), especially the ubiquitous quadratic forms that underlie KKT conditions and many optimization methods.

  2. Calculus of Vector and Matrix-Valued Functions: Views these objects as vector-objects, not sets of scalar equations. Very useful for understanding convergence of optimization algorithms and getting a new view on multivariate calculus (e.g., be able to calculate $\frac{d}{dt}A(t)B(t)$). Does not really cover Matrix Calculus, such as $\frac{d\mathbf{F}}{\mathbf{X}}$, but that is not a huge concern for me as it's mainly conventions for bookkeeping of variables.

  3. Matrix Inequalities: Pop up all the time when dealing with constraints. Also covers the extremely useful Singular Value Decomposition.

  4. (Chap 16) Positive Matrices: Perron-Frobenius Theorem -- enough said!

After that, I thought that Chapter 11 on Dynamics (to pull it together) and my own intro to Reproducing Kernel Hilbert Spaces (the conceptual world of kernel-based learning methods) would be good ways to end the course.

Again, thanks all for your input! Very much appreciated the community here.

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    $\begingroup$ Really neat course. Perhaps you could tape it so I could watch it :) Thanks for taking the time to describe your idea for this course. $\endgroup$
    – James S. Cook
    Dec 18, 2017 at 1:20

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