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I have a numerical (so to speak) understanding of linear algebra. I have read a proof-based book on linear algebra where the determinant is defined using adjoints of a row or column..., where the Cramer's rule and the method of Gauss to found an inverse matrix is presented and that sort of stuff.

However, when I took a glimpse at Lang's treatment of matrices and linear maps in chapter XIII of his algebra book, I began to notice a lot of uses of abstract algebra definitions (such as groups, homomorphism or isomorphisms - or some similar word), and to sum up, a different approach to it (at least that is what I think - note that I am just beginning my journey on higher mathematics).

My question is then whether this approach and use of different tools from abstract algebra is used to correctly and fully prove the statements of linear algebra (something like what real analysis may be in this regard to calculus) or whether these abstract algebra tools are used to actually develop new and different tools.

I am therefore asking what is the purpose of treating linear algebra with abstract algebra (if that makes sense).

If the latter option is the case, can you give me a broad view of these tools (like names and their purposes...)?

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    $\begingroup$ Linear algebra can be developed in full rigor without taking the “abstract algebra” approach you’re describing — no need to mention groups or rings, for example. If we assume the field of scalars is $\mathbb R$ or $\mathbb C$, then we don’t even need the concept of a field. See the book Linear Algebra Done Right, for example. Authors who treat linear algebra with abstract algebra do so, I would guess, because they think it’s the most illuminating way to understand the material, or the most efficient way to develop the theory. $\endgroup$
    – littleO
    Aug 5 at 20:15
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    $\begingroup$ Authors who treat linear algebra with abstract algebra do so, also because it is quite useful. As a simple example, treating eigenvalues via roots of a polynomial, so by using abstract algebra (characteristic polynomial, minimal polynomial, divisibility in a polynomial ring), is useful. $\endgroup$ Aug 5 at 20:32
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    $\begingroup$ I had two semesters of linear algebra and then two semesters of "abstract" algebra in undergrad, with the same professor. In the oral exam for Vordiplom (roughly bachelor degree) he questioned me about the Jordan normal form. The minimal polynomial of a matrix $A$ came up. He asked: "By the way, how do we know that such a minimal polynomial exists and is unique?" I said: "To be honest I don't remember how we proved it back then, but now I would say ... [he grins and gestures to me to continue] ... the map sending the variable $X$ of the polynomial ring $K[X]$ to $A$ in the matrix ring .. $\endgroup$ Aug 6 at 3:09
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    $\begingroup$ .. is an algebra homomorphism; it cannot be injective because the first has infinite, the latter has finite dimension; so it has a non-zero kernel which is an ideal. The polynomial ring is a principal ideal domain, so that kernel is generated by a non-zero element which is unique up to units. Units in $K[X]$ are scalars from the field, so if we demand the generator to be monic, it is unique. That's the minimal polynomial." The professor was pleased and later became my thesis advisor. I still don't remember how we had proved that in first year without "abstract" algebra. $\endgroup$ Aug 6 at 3:12
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    $\begingroup$ Also, related question: math.stackexchange.com/q/2061953/96384, where e.g. the answer by Eric Wofsey expands my smartass oral exam moment into saying that the whole Jordan normal form business can be translated into (and be understood through, and shed light on) module theory over polynomial rings. $\endgroup$ Aug 6 at 3:16
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You ask:

what is the purpose of treating Linear Algebra with Abstract Algebra

Reading between the lines a little bit for common motivations for this question, I think there are at least two:

  1. Why waste time learning it the way we did if we could just do it with abstract algebra?
  2. Why waste time learning it again in some weird new way when I already learned it one way?

The answer to the first one is: well, you probably wouldn't have understood it if you learned abstract algebra first then linear algebra. By reversing it and doing linear algebra first, you've softened the learning curve so that you could actually do some practical things rather quickly. Moreover, with your experience in linear algebra, you are now armed with intuition with which to more quickly learn abstract algebra.

The answer to the second one is: strictly speaking, you don't, but trust me you should give it a fair shake. But what if seeing it from that perspective made things make more sense? That is what frequently is the case. Higher mathematics layers on perspectives and permits you to develop different intuitions. Some of these are easier to learn than others.

For example, you'd never want to learn category theory before all the other branches. For one thing, you wouldn't have any examples and things really would seem like "abstract nonsense." But if one has already seen many results in several fields of mathematics, they are typically blown away when they see how category theory is "the rug that ties the room together."

Just think of how awkward it (usually) seems when a student is seeing the epsilon-delta definition of continuity. Fast forward to years later when they're taking a topology course and they're like "oh yeah, I see kind of what open sets are like (open intervals)". It would probably not work as well to teach it in the other order.

Slogan

Part of learning mathematics is layering on intuitions. Ideally these are layered in more-or-less ascending order of difficulty/complexity.

The more layers you get, usually the better you understand a subject, because you can switch between views and get more out of the picture than any single view.

Have you ever heard the anecdote about Paul Erdos (as a teenager) asking someone "How many proofs of the pythagorean theorem do you know? [...] I know 37." I have no way of knowing if he was exaggerating or kidding, but I suspect it was not far off, and it goes to show what he thought about having multiple approaches to a single thing.

Slogan

In mathematics, we frequently reverse between induction and deduction.

In the case of your example of linear algebra and abstract algebra, one can see that. Sets of square matrices can be multiplied and added kind of like integers. What else behaves like that? Let's call those things rings. (induction) Now let's deduce some things about rings...

Now, one can use that alternation to bootstrap from "simple" topics up to harder ones by learning things in a naive way (or historical way, perhaps) first, and then being armed with that experience, asking "well, knowing all that now, how could I make it easier?"

Let me leave you on one more note: you might be interested to know that the theory of representation of groups is a way to study groups (things which can be quite exotic) by reducing questions about them to linear algebra questions. Linear algebra, being relatively well-understood, is a good venue to resolve those problems. So that is a concrete situation where a hard problem is tackled by reducing it to a problem we are more familiar with.

The title question

It is, as I write this

How is Linear Algebra taught through Abstract Algebra?

I'm not sure I would say it is "taught through abstract algebra." I'd say that abstract algebra has a more general lexicon to discuss the same problems that linear algebra tackles. Many problem in linear algebra can be rephrased (and proven again) using the perspective of abstract algebra. But in the end, fields and vector spaces are very special and have many results that are not true for rings/modules in general.

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    $\begingroup$ I liked a lot your answer. It helped me understand a little better what Linear Algebra is about related with Abstract Algebra. $\endgroup$ Aug 5 at 22:12
  • $\begingroup$ Were I to have one more query it would be: Are there, then, applications of linear algebra to (for example) abstract algebra or other branches of math that could not have been reached had we developed the theory of Linear Algebra without abstract algebra? In other words, does this, in your own words, linear algebra [...] using the perspective of abstract algebra offers as new techniques to tackle other problems? $\endgroup$ Aug 5 at 22:17
  • $\begingroup$ @epsilondelta that’s a great question that I don’t think I know any good answer to. $\endgroup$
    – rschwieb
    Aug 5 at 22:31

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