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Really good stuff from what I could tell from a glance at this! "Linear Algebra Done Right" is a terrific book, though personally I favor Lax's book: "Linear Al

Really good stuff from what I could tell from a glance at this! "Linear Algebra Done Right" is a terrific book, though personally I favor Lax's book: "Linear Algebra and its Applications". I have not studied Terence Tao's notes (which I obtained from AMS's link for "open math notes") https://www.ams.org/open-math-notes/omn-view-listing?listingId=110650 https://www.ams.org/open-math-notes/omn-view-listing?listing.... Based on what I know of Tao's writing in general, this should be fun. Note that these references have a more "abstract" viewpoint (e.g focusing on "coordinate-free" methods, such as linear operators as opposed to matries as the fundamental object) than that advocated and taught by Strang. I do not know in the end which is better for an absolute beginner. All I can guarantee is that the abstract viewpoint is definitely needed and far more useful for anyone using math beyond its basics. This includes (but is not limited to) machine learning and optimization.

geezerjay 9y ago
> Really good stuff from what I could tell from a glance at this! I agree. I'm really glad this was only posted on a weekend because now I'll have to spend the day watching all the videos. Awesome stuff.
js8 9y ago
I agree with you. My teacher at Czech Technical University had a similar, abstract approach to linear algebra - also used linear operators all the way, matrices were only mentioned in passing (and in exercises). It was a tough course for 1st year freshmen (almost as difficult as mathematical analysis), due to its abstract nature, but I was grateful for that in the end.
woopwoop 9y ago
I haven't read Lax's linear algebra book, but I fell in love with his functional analysis book in the summer before I began grad school. His notes on hyperbolic pde are also great. Lax is of course a great mathematician, but he is also an incredibly lucid writer. He doesn't make anything more difficult than it needs to be. I would recommend reading anything written by him that you can get your hands on.
- tanderson92 9y ago
  There are three problems with Lax's book, from my experience (I used Lax in undergrad and Axler later and I would have gotten more from each if the order were reversed). 1) Lax leaves significant results to the problems, and the result is that there is little explanation of their importance. 2) Lax prefers short, mathematically impressive proofs which leverage the mathematics he has developed in the book to that point. But the proofs do not help in understanding. 3) Lax relies heavily on determinants and characteristic polynomials to prove major results. Axler deliberately avoids them, and he says why in the introduction too! Axler and Lax have two diametrically opposed approaches to linear algebra pedagogy. Neither is "wrong" but in my experience Axler's is much better for learning and Lax's is better for a deeper understanding (and for a different approach).
- gajjanag 9y ago
  Thanks a lot for this endorsement of his functional analysis book. I never got around to understanding functional analysis properly; mainly because I don't work on PDE's/control theory that would have used the ideas more heavily. Added to my todo list. Not really on topic, but I would like to use this opportunity to express gratitude to the HN community - although this forum is mostly tech-centric, it is wonderful to see things like this post get interest and discussion.
  - tanderson92 9y ago
    Thank you for saying so! It's good to not feel left out. There are plenty of mathematicians and applied mathematicians here, too. We just do not comment on the JS/Rust discussions :)
Tomte 9y ago
Oh great! So there is "Linear Algebra and Its Applications" by Lax. And "Linear Algebra and its Applications" by Lay. At least "its" is capitalized differently. ;-)
btilly 9y ago
I believe that the abstract viewpoint should come first, but should be tied very, very thoroughly to the concrete calculations. The construction of a matrix to represent a linear operator in two concrete coordinate systems should be completely understood, forwards and backwards. The point of the abstraction not being to have an abstraction, but to give intuition to the otherwise pointless-seeming calculations.
jcranberry 9y ago
I'm not a particularly advanced student, but I took an introductory class using Lax and then later material with an algebraist professor and her notes. I struggled with both, but I found it difficult to understand Lax's material until I was introduced to Galois/Invariant Theory and it all became much more clear. With the notes I used in the latter class I felt I gained a very deep understanding in return for my efforts, especially because they went into Complexification and bilinear/sesquilinear forms which gave me a really strong intuition on the geometry of matrix representations both real and complex. If I remember correctly, Lax didn't really focus on those concepts Not that Lax is a bad book but I feel like Lax's Linear Algebrs is actually a sneaky way of familiarizing the reader with formalisms they'll encounter in later, more abstract material, so maybe better for students who are already comfortable with a certain amount of abstraction so they're able to gain enough to leverage it later but not necessarily for a truly introductory class.

geezerjay 9y ago

> Really good stuff from what I could tell from a glance at this! I agree. I'm really glad this was only posted on a weekend because now I'll have to spend the day watching all the videos. Awesome stuff.

js8 9y ago

I agree with you. My teacher at Czech Technical University had a similar, abstract approach to linear algebra - also used linear operators all the way, matrices were only mentioned in passing (and in exercises). It was a tough course for 1st year freshmen (almost as difficult as mathematical analysis), due to its abstract nature, but I was grateful for that in the end.

woopwoop 9y ago

I haven't read Lax's linear algebra book, but I fell in love with his functional analysis book in the summer before I began grad school. His notes on hyperbolic pde are also great. Lax is of course a great mathematician, but he is also an incredibly lucid writer. He doesn't make anything more difficult than it needs to be. I would recommend reading anything written by him that you can get your hands on.

tanderson92 9y ago
There are three problems with Lax's book, from my experience (I used Lax in undergrad and Axler later and I would have gotten more from each if the order were reversed). 1) Lax leaves significant results to the problems, and the result is that there is little explanation of their importance. 2) Lax prefers short, mathematically impressive proofs which leverage the mathematics he has developed in the book to that point. But the proofs do not help in understanding. 3) Lax relies heavily on determinants and characteristic polynomials to prove major results. Axler deliberately avoids them, and he says why in the introduction too! Axler and Lax have two diametrically opposed approaches to linear algebra pedagogy. Neither is "wrong" but in my experience Axler's is much better for learning and Lax's is better for a deeper understanding (and for a different approach).
gajjanag 9y ago
Thanks a lot for this endorsement of his functional analysis book. I never got around to understanding functional analysis properly; mainly because I don't work on PDE's/control theory that would have used the ideas more heavily. Added to my todo list. Not really on topic, but I would like to use this opportunity to express gratitude to the HN community - although this forum is mostly tech-centric, it is wonderful to see things like this post get interest and discussion.
- tanderson92 9y ago
  Thank you for saying so! It's good to not feel left out. There are plenty of mathematicians and applied mathematicians here, too. We just do not comment on the JS/Rust discussions :)

Tomte 9y ago

Oh great! So there is "Linear Algebra and Its Applications" by Lax. And "Linear Algebra and its Applications" by Lay. At least "its" is capitalized differently. ;-)

btilly 9y ago

I believe that the abstract viewpoint should come first, but should be tied very, very thoroughly to the concrete calculations. The construction of a matrix to represent a linear operator in two concrete coordinate systems should be completely understood, forwards and backwards. The point of the abstraction not being to have an abstraction, but to give intuition to the otherwise pointless-seeming calculations.

jcranberry 9y ago

I'm not a particularly advanced student, but I took an introductory class using Lax and then later material with an algebraist professor and her notes. I struggled with both, but I found it difficult to understand Lax's material until I was introduced to Galois/Invariant Theory and it all became much more clear. With the notes I used in the latter class I felt I gained a very deep understanding in return for my efforts, especially because they went into Complexification and bilinear/sesquilinear forms which gave me a really strong intuition on the geometry of matrix representations both real and complex. If I remember correctly, Lax didn't really focus on those concepts Not that Lax is a bad book but I feel like Lax's Linear Algebrs is actually a sneaky way of familiarizing the reader with formalisms they'll encounter in later, more abstract material, so maybe better for students who are already comfortable with a certain amount of abstraction so they're able to gain enough to leverage it later but not necessarily for a truly introductory class.