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Comment by doppioandante | Hacker News Reader

There's a introductory book that I would add to the "generalist" list, that is 'Linear Algebra Done Wrong' by Sergei Treil. It can be found freely here: https://www.math.brown.edu/~treil/papers/LADW/LADW.html

I've used it to prepare my linear algebra exam in an italian university (which usually emphasize theory over practice, even in engineering courses), and I've loved it. There are few exercises but they're to the point. Complex inner product, self-adjoint operators etc. are treated. I find the notation used in there particularly easy to understand, and proofs are elegant but very easy to follow.

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I have skimmed some books about Linear Algebra, both in Italian and in English, and my favourite so far is "Geometria e Algebra Lineare" by Enrico Schlesinger.

The book is very well written but sadly it won't be enough to pass a typical italian Linear Algebra exam. This is because, in my opinion, examples are too vague or too "abstract": when you're first learning such subject, it would be better to have layman terms, that is simple examples with simple numbers. More complex examples might be okay too, but simple examples should come first.

Reference: http://www.zanichelli.it/ricerca/prodotti/algebra-lineare-e-...

Marco Abate's "Geometria" was also quite entertaining.

My favourite linear algebra book in italian is "Geometria e Algebra Lineare" by Abbena, Finno and Giannella. Really great book, with a wealth of exercises and extensive material.

I can also suggest the youtube lectures[0] by Francesco Bottacin, who actually was the one that eventually did my oral examination.

He's also written two books based on the course, the one with exercises[1] is very solid and I think you can pass the written exam by just doing it whole. They told me he is a wonderful lecturer. Pity that all this material is in italian.

[0] https://www.youtube.com/channel/UCpJGeVBfmf-6S3neeImAw1w/fee...

[1] https://www.amazon.it/Esercizi-algebra-geometria-Francesco-B...

You'd probably like Strang. He's an mit professor who shared his linear algebra course lectures. They're excellent.

http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-...

A one-edition back version of his book _Introduction to Linear Algebra_ is available for under $30.

https://www.amazon.com/Introduction-Linear-Algebra-Gilbert-S...

Unfortunately that book was too lightweight on theory to be of use to me, but it's certainly a very good book. I've recently started Strang's more advanced book on computational linear algebra/numerical analysis and it's pretty good though.

I actually don't like the Strang book and his MIT course.

I watched 6 or 7 of his lessons, and to be honest, I think they're nice if you already know the subject.

I have skimmed both an italian version of his book and an english one, still didn't like it.

The nice thing about the Schlesinger book is that it builds all of the concepts basically from scratch. Euclid’s Elements are quoted too, in the introduction.

Too bad it has poor examples.

That was not my experience -- I was utterly confused by my college linear algebra class which was taught as a bag of techniques. I couldn't figure out why anyone would care or what you would do with any of it. And I definitely didn't get the connections to the geometry that Strang goes over in Ch4 iirc.

Strang taught me linear algebra. His course -- and I read a couple other books afterwards -- is the only reason I understand it. Not only the how of the techniques, but why and what they do.

I don't suppose there's an english translation of the Schlesinger book? Google says no.

I was on the fence about that one and ultimately took it off the list. It seems to be fairly numerical, sort of in defiance to Axler's "Done Right" book. But then as a numerical book how does it compare with the others in that category, or with the old Russian books? I ask quite honestly because I haven't gone through Treil's book. Does it hold its own?

Actually, I think it's very similar to Axler's in spirit. Not numerical at all (there is a description of the min squares problem, but that's it as far as I remember). Matrices are used more explicitely than in Axler, but only when necessary, because the author claims he's an operator theorist so the focus is on linear operators. Determinants are introduced as multi-linear functionals and are not avoided as in Axler. Supporting the fact that the book is quite theoretical, this is what you can find in the last chapters:

Chapter 6. Structure of operators in inner product spaces. Chapter 7. Bilinear and quadratic forms Chapter 8. Dual spaces and tensors Chapter 9. Advanced spectral theory

Unfortunately I'm not familiar with the Russian books.

Linear Algebra Done Wrong is quoted at many places with another book - Linear Algebra Done Right. Did you study other book - any thoughts?

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