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Comment by tadhgds | Hacker News Reader

I can't say whether or not it is the standard approach but I do know that it is very common in many countries to teach a linear algebra course that is heavy on matrix operations, that you can come away believing that linear algebra is somehow _about_ matrices and their operations. I know many in my university class seemed to believe that.

A book I enjoyed is Axler's Linear Algebra Done Right[0], in which, if I remember correctly, doesn't contain a single matrix.

[0]https://zhangyk8.github.io/teaching/file_spring2018/linear_a...

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I've recently started going through Axler carefully and doing the problems, a quarantine activity I guess, and have been enjoying it. I actually learned about this book on an older HN post.

It does have plenty of matrices. The main thing it really does is avoid determinants until the very end. The determinant is certainly something I remember learning as a kind of rote operation, without really understanding any intuition behind why you'd multiply and add these numbers in this particular way. I still feel lacking in "feel" here, which is why I suppose I'm going through Axler now.

Yeah, math professor here, this drove me crazy.

For example, I remember looking at the linear algebra book my department had used previously. Early on, it introduced the concept of the transpose of a matrix:

https://en.wikipedia.org/wiki/Transpose

Superficially, it looks like something good to introduce. It is fodder for easy homework exercises, and there is a satisfyingly long list of formal properties satisfied.

But why? What does the transpose mean? For what sort of problem would you want to compute it?

There are good answers to these questions (see the "transpose of a linear map" section of the Wikipedia article I linked), but they are not easy for a beginner to the subject to appreciate.

IMO Axler's book should be read either during or after you take an introductory course on Linear Algebra.

> You are probably about to begin your second exposure to linear algebra. Unlike your first brush with the subject, which probably emphasized Euclidean spaces and matrices, this encounter will focus on abstract vector spaces and linear maps.

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