What's next? $\text{Hom}(U,V)$, tensor product and so on

I don't think this question is as soft as soft-question tag, but if I not how to tag it, I would not be asking this question:

I have taken second year linear algebra (Axler's Linear Algebra Done Right), but I still find much trouble understanding $\text{Hom}(U,V)$, like what is its basis; and no idea about tensor product.

So should I consider "a third year linear algebra"? Though there is not really a course named so - but is there a course or book I should read after Axler? Or I can learn more about these in some other course, like functional analysis? Is Representation theory related/necessary at this level?

• Absolutely lecit question and very important topic. Learn Hom and the tensor product $\otimes$ in a parallel way: they are deeply interlaced. I suggest you to read Jacobson's "Basic Algebra I-II" or Lang's "Algebra": they are both nice books. In my opinion Jacobson's is really a good text. – Avitus Jun 18 '13 at 18:28
• Good point, thanks @Avitus. So, more algebra - thanks! Also, what does lecit question mean... @_@ – 1LiterTears Jun 18 '13 at 18:33
• Sometimes soft and even not so soft questions are hit by the "censorship" of other Math.SE users or generate not so interesting debate between them. This is not the case :-) – Avitus Jun 18 '13 at 18:38
• That's very kind of you to say so, thanks @Avitus. – 1LiterTears Jun 18 '13 at 18:40

It can be often illuminating to forget some of the technology and go back to a simpler time. Axler's text is a favourite of mine. If you can do all the problems in that book, then you know your linear algebra, let me assure you.

Regarding $Hom$...

You have two vector spaces $U,V$ (lets say they're finite dimensional). In plain english, $Hom(U,V)$ is the set of all (linear) maps from $U$ to $V$. We're in a finite dimensional case so lets go ahead and pick bases for $U$ and $V$ so they each look like $k^n$ for (possibly different) $n$. Think of them as column vectors.

So, how do you map one set of column vectors to another: you use matrices! You know how to scale matrices, and add matrices. In other words, $Hom(U,V)$ has just been given the structure of a vector space all on its own. Its a big vector space consisting of matrices of the appropriate size (so multiplying them with column vectors from $U$ makes sense) and one basis for instance is given by putting $0$ everywhere in the matrix except in one spot.

You can write this down in fancier ways without choosing bases too: Given maps $\phi$ and $\psi$ from $U$ to $V$, we can (pointwise) add and scale them and we get new linear maps. It is a good exercise to figure out what the basis I described above in terms of matrices looks like without explicitly thinking about the matrices. It is going to be a distinguished linear transformation. Figure out what it is!

Another great exercise is to set $V = k$ and understand $U^*$, the dual vector space. What do the above matrices look like in this case? Can you see for yourself that $U\cong U^*$ once you picked a basis?

Tensor products are somewhat more subtle. I would suggest trying to understand them in terms of what they do''. The tensor product of $U$ and $V$ is built to be the most general machine that can handle linear maps from $V\times U$, that are in fact bilinear (linear in each variable). Sit on this idea for a while. Then look at how a tensor product is defined in terms of generators and relations.

Once you have, try and make sense of: $k^n\otimes k^m \cong k^{mn}$.

• Thanks so much for your generous guidance. First a quick question - are you suggesting to redo exercise on Axler instead of move to new materials? – 1LiterTears Jun 18 '13 at 18:10
• Well, perhaps not every single one. But at least until you get to the point that you arent overwhelmed by the language. It is difficult to learn things like functional analysis or representation theory without being fully comfortable with vector spaces without distinguished bases. Axler does a particularly good job of getting you to do this. Even if you just redo exercises you've already done, it would be helpful to make sure you actually know the inner workings of the big machines. – Dhruv Ranganathan Jun 18 '13 at 18:39
• Thanks Dhruv! I guess I followed your guidance until the very last - could you point out how to prove that $k^n\otimes k^m \cong k^{mn}$? Thanks.! – 1LiterTears Jun 18 '13 at 21:22