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Just finished "Linear Algebra Done Right" and linear algebra just isn't 'clicking' for me.

18 Answers

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I was kind of the same when I first learned LA. Without applications it can seem a bit dry and pointless. You will start seeing it everywhere in other areas of math. I wouldn't sweat it right now.
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The big picture is that linear algebra is relatively easy to understand but also powerful and ubiquitous in its applications (both mathematical and practical). A proof-focused course may not have given you a good sense of just how useful and ubiquitous it is.

A common recipe for solving problems is to try to translate them into the language of linear algebra. If you can turn a problem into linear algebra then you’re essentially guaranteed to understand how you should approach solving it, and this strategy works so often that some people learn to use it as their first approach for almost everything.

I think that linear algebra is a bit like probability in the sense that one of the biggest challenges with it is learning how to translate concrete problems into the relevant abstractions. This is a skill that is acquired primarily through practice, and it’s one that might easily be neglected in a course that focuses on doing proofs about linear algebra in the language of linear algebra.
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If you have any interest in numerical analysis at all, Numerical Linear Algebra by Trefethen and Bau is an amazing book for where you're at.
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Imo linear algebra is one of those things that makes more sense in hindsight. As you go forward in math it will start popping up pretty much everywhere. I don't think Axler really spends much time on applications unfortunately. The book by Friedberg etc has a lot of applications that might put it in perspective.
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I believe it _is_ supposed to be like this when you encounter it for the first time. Felt the same to me too. It's one of those foundational topics that pervade a really vast terrain in maths, yet on their own feel almost useless and dry. That's because it's an abstraction, a generalization of some useful concepts that pop up in many different places. For example, if you're going to take multivariable calc you'll see it pop up a bit. If you take differential equations, you'll see it pop up quite a bit. If you take machine learning, that's basically entirely applied calc and lin alg. Courses further up the ladder, e.g. probability theory, abstract algebra, differential geometry etc, continue becoming more and more dependent on linear algebra. If you take physics courses, quantum mechanics, special and general relativity, continuum mechanics etc make heavy use of lin alg. Heck, it's not just useful in pure maths and physics - basically any field that involves mathematical modelling will depend highly on linear algebra. The specifics often don't matter but it's very very useful (and necessary) to have a high-level understanding. Also many of the techniques you learnt in this course will eventually become a permanent part of your arsenal.

The basic takeaway right now should be how linear transformations work, how we can represent them with matrices, how the same transformation looks different in different bases and that transformations will in general have a preferred basis where they look really simple. I studied a few chapters of Axler several years ago and I believe the book does a good job of introducing these things. Nevertheless, I'd say have a go at watching _3blue1brown_'s YouTube playlist called _Essence of Linear Algebra_. It will, hopefully, make a lot of things click for you.
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I’ll second Terefethen and Bau.  When you’re using it to solve problems that are more realistic it starts to make more sense.  However, that book will take some work.  Matlab helps quite a bit.
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There is a YouTube channel called 3blue1brown that did a series on the intuition of linear algebra.

Excellent and very digestible content which I would suggest any novice of the field to take a look.
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I think *Linear Algebra Done Right* is notorious for being a love-it-or-hate-it book. It works well for some people and terribly for others.

My linear algebra course used the Friedberg, Insel, and Spence text, which I would rate as "fine."

However, what really clicked for me was taking Abstract Algebra afterwards. I found that LA was a tough course to learn proof techniques alongside the topical material, but AA was much more straightforward, so I was able to get comfortable with proofs and the material at the same time. I'm very confident that if I were to go back through a LA course, I'd get much more out of the material now than I did the first time around.
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It’s used a lot in statistics. You can’t understand linear regression thoroughly without linear algebra. Any area which uses big data or is data science related will require linear algebra. I guess pick up an area of math where it becomes useful. For example I was able to appreciate the power of eigenvectors in principal component analysis.
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I believe Linear Algebra is best taught according to applications and then filling in the gaps. In my LA course (which was more computer science focused), we went through various applications like audio/image compression, heat modeling, etc. Machine Learning wasn't as big back then but that would also be a good next step. These would all be fairly simple/basic concepts, but at least they provide the grounding.

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