Insights How to Self-study Algebra: Linear Algebra - Comments

[micromass]

micromass submitted a new PF Insights post

How to Self-study Algebra: Linear Algebra

Continue reading the Original PF Insights Post.

 

[ibkev]

Great list - thanks for putting this together!
The author of Linear and Geometric Algebra, Alan Macdonald, has put together some youtube videos in support of his book.

https://www.youtube.com/c/AlanMacdonald1/playlists

 

[ibkev]

Anyone intending to tackle both the Linear Algebra Insight and the Intro Analysis Insight, will probably notice that there is some overlap between the two. Micromass was kind enough to provide an efficient way to navigate through them, which he gave permission to repost here:

"So if you're doing both of them, then I would recommend:

Do Bloch Analysis and MacDonald in parallel.
Then after Bloch do Hubbard, and after MacDonald do Axler.

This way you'll get everything without too much repetition. MacDonald will teach you the basics of LA (vector spaces, linear transformations), but will also do geometric algebra. Hubbard will repeat the basics but not from a point of view of analysis. And Axler will do things in the most rigorous light. Avoiding determinants in Axler is not a problem since Hubbard and MacDonald cover those. What do you think? It is possible to do Treil instead of Axler if you prefer Treil, but it's really up to you."

 

[ibkev]

One thing about the Macdonald book is how surprisingly small it is (204 pages) for the amount of content it seems to cover. This is mainly for 2 reasons: (1) it handles worked exercises in a cool way and (2) he doesn't devote space to learning what he calls algorithms (e.g. the mechanistic cookbook recipe for row reduction, etc.)

Regarding worked exercises, the trick is he has you do them! Almost every page has a couple of small exercises that relate to the text you just read. They really make you engage with the content as you go in a neat way that I haven't seen before. Sometimes you'll want a little scratch pad and a pencil to work it out and other times it'll be something simple that you can work out in your head like "what happens if you set t = 0 or 1?" and then you have an aha moment as you realize it simplifies to something you've seen before. This is quite rewarding as opposed to being given the same information in a paragraph.

Regarding algorithms, an example is matrix inversion - he goes through the concept and applications of it, thereafter using it throughout the book but he does not devote space to building up the detailed recipe for mechanistically computing one by hand. Same goes for row reduction, determinants, eigenstuff, etc. In the Preface he argues that the recipes are not needed for theoretical development, and no one solves them by hand anymore anyways except as exercises in Linear Algebra textbooks.

 

[Calaver]

Thank you micromass for the second insight this month that meets me where I'm at (the other was the one about advanced mathematics for high school students)!

As someone who is currently attempting to self-study linear algebra (albeit very slowly), this post helped me to see what I can expect to know once I'm done (by what micromass says the other books cover), especially since I'm using a fairly "mathematically pure" book to study from that doesn't give much motivation the whole subject.

Since it didn't make it on micromass' list, I'll go ahead and mention the book I'm using for anyone who is interested. It's by Shilov, just called "https://www.amazon.com/dp/048663518X/?tag=pfamazon01-20." It starts from determinants, which I hear is a different approach than most books take. The second chapter is on linear spaces though, so it seems that it still gets to "the point" rather quickly. I find it excellent, and perhaps it should be considered for someone who is looking into the subject.

Although I will point out that I have no prior experience with linear algebra and have looked at no other texts, so please take micromass' textbook advice over mine (if only this post had come out before I started to get into the book ).

 

[micromass]

Thank you micromass for the second insight this month that meets me where I'm at (the other was the one about advanced mathematics for high school students)!

As someone who is currently attempting to self-study linear algebra (albeit very slowly), this post helped me to see what I can expect to know once I'm done (by what micromass says the other books cover), especially since I'm using a fairly "mathematically pure" book to study from that doesn't give much motivation the whole subject.

Since it didn't make it on micromass' list, I'll go ahead and mention the book I'm using for anyone who is interested. It's by Shilov, just called "https://www.amazon.com/dp/048663518X/?tag=pfamazon01-20." It starts from determinants, which I hear is a different approach than most books take. The second chapter is on linear spaces though, so it seems that it still gets to "the point" rather quickly. I find it excellent, and perhaps it should be considered for someone who is looking into the subject.

Although I will point out that I have no prior experience with linear algebra and have looked at no other texts, so please take micromass' textbook advice over mine (if only this post had come out before I started to get into the book ).

Don't worry, Shilov is an excellent book. You really can't go wrong with it. I personally wouldn't recommend it because it does determinants in the beginning, which I find a fairly unintuitive and perhaps too abstract approach. Also, it never really says what a determinant is geometrically (as far as I recall). But if you like it, it's a nice book.

 

[CynicusRex]

This surely fits the list: https://www.lem.ma/
Made by: http://drexel.edu/coas/faculty-research/faculty-directory/PavelGrinfeld/

 

[Thomaz]

what are your thoughts on Kenneth M Hoffman's Linear Algebra textbook?

 

[mathwonk]

I think Hoffman and Kunze is a bit abstract for most beginners, having been aimed at a junior level math major at MIT.

after reading some introductory book you might possibly gain something from my notes. In the first 3 pages I summarize the entire content of the most advanced parts of most books, the jordan and rational canonical normal forms. it is free of course. i also like shilov very much, but have not really worked through it. a lot of people, including me, like friedberg insel and spence.

http://alpha.math.uga.edu/%7Eroy/laprimexp.pdf