How Can I Keep Advancing in Linear Algebra?

Question

I recently graduated with a bachelor's in computer science and a minor in mathematics. Since I won't be enrolled in any more math classes, I want to continue learning and honing my skills by myself. Currently, I am reteaching myself Real Analysis through a textbook. It is interesting, but linear algebra has been, by far, the most interesting section of mathematics for me.

After taking my first linear algebra class which was heavy in theory, I took an applied linear algebra class. Unfortunately, this class didn't teach me too much - it mainly just reiterated the basic concepts from the first class I took. I assumed it would pertain more to things like machine learning since it was applied. But I wasn't exposed to too much new material.

I then took an intro-level machine learning class and that really made me see the beauty of linear algebra. After this, I realized I wanted to continue learning about this area of mathematics.

Where can I go from here? What is the next step? I'm sure there's a ton more to learn. I will let it be known that I am no rockstar mathematician. I often spend a lot of time pondering certain problems that many could probably figure out very quickly to ensure I have a very strong grasp on what's happening rather than settling with a vague or just solid understanding. However, as I am proving to myself right now, I do have the discipline to teach myself out of a textbook. I hope this is possible in higher maths, especially for someone of my skill level. I am willing to give it a shot and I am very interested.

I'm mainly looking to be steered in the right direction, whether that means specific areas of math and/or textbooks. All advice is appreciated.

UPDATE

My first linear algebra class was taught from the book Linear Algebra with Applications by Steven Leon. To clarify what I meant by the class being heavy in theory, I mean our homework and exam problems would be centered around proofs. This is in contrast to my applied linear algebra class, that much more geared around computational problems. That class was taught out of the book Matrix Analysis and Applied Linear Algebra by Carl D. Meyer.

As for what I'm interested in, it's kind of hard to say. I'm not fully knowledgable on all of what maths and linear algebra have to offer, but all that I've learned I've enjoyed. Another area of math I thoroughly enjoyed learning about was abstract algebra if that helps in suggesting what other things I might enjoy.

Answer 1

Judging from what you wrote about "relearning" real analysis, it seems you're looking for a rigorous book on linear algebra and you don't mind if there's overlap with what you've done before.

For people who are interested in math for its own sake, I think it makes sense to learn abstract and linear algebra together. A good book for this is Algebra by Artin, which is geared towards undergraduates with a relatively high (but not insanely high) level of ability. It's best if you're relatively comfortable with vector geometry before starting this text, but this isn't strictly necessary.

(Added: Algebra by Godement covers a bit less than Artin but has harder problems. It also has a much more formal tone.)

For books that would be suitable for an average math major, a couple of reasonable choices would be Introduction to Linear Algebra by Serge Lang, which is concise and has a good introductory chapter on vector geometry, and Linear Algebra by Friedberg, Insel and Spence, which is quite wordy and introduces proofs gently. (Added: Given your background, you may find this last book slow going. It's only slightly above the level of the book by Leon.)

These last two might not be the best books for people who already have a good deal of what's called "mathematical maturity." Some concise treatments of linear algebra for people already comfortable with abstract math would include Lectures on Linear Algebra by Gelfand, Finite-Dimensional Vector Spaces by Halmos, and Linear Algebra and Geometry by Manin and Kostrikin.

Edit. If you feel what you really need is a book of solved problems to help you see how to apply the theory you already know, two of the best are Problems in Linear Algebra by Proskuryakov and Problems in Higher Algebra by Faddeev and Sominsky.

Once you're proficient with the basic theory, given your CS inclinations, you may want to turn your attention to computational linear algebra. Matrix Computations by Golub and Van Loan has a good bibiography in this area.