Why is linear algebra so big and scary?

>linear algebra
>big and scary

Yeah, I remember my sophomore year of college.

Fucking weebs trying to rewrite history.

But yeah, linear algebra is cozy af and you'd do well to learn the concepts and not just plug and chug.

The whole point of linear algebra is that it's really fucking easy. That's why it plays such a fundamental role in all mathematics that matters.

>linear algebra building up to hilbert's third problem
I don't think that's so easy user

Which linear-algebraic part do you find non-easy?

Linear algebra is definitely easy compared to other areas of math
Most of the time in linear algebra, if something feels 'right' or intuitive, it usually is, and the proof is usually not very complicated. Once you get past the matrix machinery and start dealing with the actual algebra part of it, the whole thing becomes clearer, and the course gets progressively easier.

Compare to analysis, where if you feel that some property holds intuitively, then fuck you and your children because there's some obscure-as-fuck counterexample to completely smash your intuition, and the construction of said counterexample is a series of steps longer than Kanye West's monster dong.

Sesquilinear/sylvester/free abelian groups.
The theory isn't easy.

The scissor congruent section had no such machinery, and neither did tensor products.

Ah, ok, if you articficially expand the notion of linear algebra to not necessarily semisimple abelian categories, then I agree. But it's a pretty lazy way to 'win' an argument. Also, if you happen to use the phrase linear algebra in real life you might want to clarify yourself to prevent confusion. People might expect you to mean linear algebra instead of 'linear algebra'.

>k-linear maps are easy
>field extensions are easy
>representations are easy
guys no

Kanye west has a large penis? Sauce?

But it is linear algebra - first year smith normal form extends itself to finding a new basis for unimodular matrices of finitely generated abelian groups.

Sesquilinear maps and Sylvester's theorem on the signature of a matrix isn't in this "abelian category" that you speak of.

No buddy. It's not.

Question 1 from a homework in one of my courses. Entirely linear algebra. I found it quite challenging, but very fun. I got to use cayley-hamilton :3

How is it not? It's literally in a linear algebra module starting with Jordan normal form.

Never claimed they were. But they are auxilliary structures and games you play in these semisimple categories. All I said is that higher dimensional abelian categories aren't considered part of linear algebra. They tend to be where you do homological algebra. Which, in contrast to linear algebra, can be somewhat difficult and non-trivial.

I'll put it in as simple a language as I possibly can:
Short exact sequences of abelian groups do not necessarily split.

That's pretty nice user.

They are direct extensions of your usual bilinear maps which you DO definitely meet in linear algebra.
But yes, some may not consider abelian groups to be part of linear algebra.
JNFs need not split either.

My professor has such neat handwriting; his sigma for the Riemann sum is done so meticulously.

If you don't mind, I'm going to steal that image to work on once finals are over.

>JNFs need not split either
What? My point was that Z is 1-dimensional. What does the notion of splitting for JNFs have to do with anything?

Only ODEs and PDEs have anything written by hand for us so we don't really get to see the "true" handwriting of lecturers (as opposed to at the board).

I assumed you were talking about representations with n x n matrices corresponding to them (which would have been a very weird restriction).
But yes, deeper abelian work is NOT linear algebra, but its introduction shows itself in early linear algebra. How else can you fill a 10 week course on linear algebra? There's not really much content to go through.

At my university linear algebra is a first semester course, just over reals and complex numbers. By the end I think the students know about eigenvalues. No normal forms, not even tensor products.
It's a pretty shitty situation, because since it's their first exposure to somewhat abstract objects we have to take it slow. Ultimately we end up having to spend time during courses like representation theory to teach them linear algebra. It's such a fucking waste.

>It's such a fucking waste.
I agree, I'm sorry to hear that user.
Eigenvalues are something that are learnt in highschool and if you haven't seen them, then you better learn them in the first term since they are seen all over the place in various courses (e.g. a text/lecturer may say "recall from pre-university...").
The only field that we do not consider is of characteristic two, since then plenty of theorems on isomorphisms between bilinear, quadratic and symmetric forms fail. Instead it gets left as a potential second year essay topic (yes, we have to write an essay for second year, which is not too bad as it forces you to learn LaTeX).

Do you think you could just throw a prerequisite text and them and expect them to know it before your course starts? It'd save a lot of fucking time.

Asking students to do something before the course starts does not sound realistic, at least not where I'm from. Also, eigenvalues in highschool? It sounds like you come from a wonderful academic environment and I hope you appreciate that.

>k-linear maps not easy
>field extensions not easy
>representations not easy
i will admit that representations took longer than usual to understand, but multilinear maps and field extensions gave me no trouble and i'm an idiot

also, are field extensions and representations even part of linear algebra?
i know what representations are and what they're used for, but i didn't learn about them until near the end of my first abstract algebra course
and field extensions were never used in my linear algebra course, which was quite involved and rigorous

>Asking students to do something before the course starts does not sound realistic, at least not where I'm from.
Is your course a first or second term/semester course? If it's first you could try and ask it to be done in the summer... but if it's not something you traditionally do there, it would indeed be difficult.

>It sounds like you come from a wonderful academic environment and I hope you appreciate that.
I didn't, my highschool went through 5 headteachers whilst I was there and had to revamp itself many times to try to impress the education authorities (we made it into the papers for receiving the worst grade possible).

Field extensions and representation theory are not traditionally thought of as linear algebraic topics.

Field extensions were covered with characteristic polynomials and Cayley-Hamilton for us.
Representations were in the last chapter of a 3 chapter course.
On the other hand, our half course abstract algebra didn't touch representations at all.
As for multilinear maps, it had very heavy differentiation theory (Gateaux and Frechet derivatives) attached with it, which isn't strictly linear algebra at all.

jesus christ what a clusterfuck
not only could this have been said in a much clearer, more intuitive way, the typesetting is awful
who did this?

Google Daan Krammer, he uses his own font packages and everything. A lot of his stylistic choices are a bit weird.

>also, are field extensions and representations even part of linear algebra?
No. I think his logic is that if a vector space comes up in studying a subject, it must be linear algebra. However, this would mean all of math is just linear algebra.

Or that it was part of a course called "Linear Algebra".

You would discuss lots of commutative algebra results in a number theory class. I don't think that would make modules a part of number theory.

Modules aren't a part of number theory?
Apparently I don't know how to define parts of maths.