Suppose $T: V \to V$ is a linear transformation and $\mathbf{v} \in V$ such that "$T^4(\mathbf{v}) = 0$ while $T^{3}(\mathbf{v}) \neq 0$." The question states to prove that $\{\mathbf{v}, T(\mathbf{v}), T^2(\mathbf{v}), T^3(\mathbf{v})\}$ is linearly independent.
So I looked back at a linear algebra book I have.
I know that we must prove that $a_1\mathbf{v} + a_2T(\mathbf{v}) + a_3T^2(\mathbf{v}) + a_4T^3(\mathbf{v}) = 0 \implies a_1 = a_2 = a_3 = a_4 = 0$.
Wow, I couldn't have thought of that idea. So we have $a_1T^3(\mathbf{v}) + a_2T^4(\mathbf{v}) + a_3T^{5}(\mathbf{v}) + a_4T^{6}(\mathbf{v}) = 0$ using linearity.
I need to know if I'm doing this exercise correctly:
$f(z)$ has only one singularity in $z=0$, and it's a pole of order 7
$f(z)=-f(-z)$
$f(z)$ is analitic in $z=\infty$
$g(z)$ is analitic everywhere except of $z=\infty$, where it has a pole of order 7
I need to calculate $Res[f.g]$ in $z=0$
...
@Juan: I commented on your post. For starters, you're not thinking correctly about how to multiply two series together. Think about multiplying polynomials.
His story is also amazing. If you read his wikipedia entry, you would think it was a smooth (no pun intended) trajectory from one high profile position to the next. But it wasn't like that at all, it seems :)
I read a paper of his (and gompf, morrison, walker) recently, where they proposed some ideas for studying the smooth 4d Poincare conjecture. Shame that the thing they spent most of their computation time on was proved to not possibly work. :P
Well, they didn't know it wasn't how not to prove it yet, @Ted. And the fundamental idea they were doing is still workable. Just the invariant they were using to verify that... doesn't.
The reason I didn't end up doing arithmetic geometry was because nobody here quite did what I was interested in, @AlexWertheim. Khare is really into modular representation theory, and I'm not. (I was going to say something about Hida but then I remembered I have no comprehension of what he does. I think I tried to say something about how much I liked ____ on my application, but I had honestly no idea.)
Interesting @Mike. It's honestly one of my concerns. I don't know enough number theory quite yet to know whether or not my future interests would match up well, and despite being well known, UCLA's number theory department is pretty small, no?
Bill Duke does analytic number theory, which isn't much my cup of tea. It seems most number theory students are studying with Hida or Khare.
In some sense, sure, but it's vibrant. And yes, that's correct, most are studying with them. You'd have to ask other number theory students - I'm a bad source.
I probably would have enjoyed it had I gone that route, but it didn't have the immediate appeal that, say, Poonen's or Bhargava's or Silverman's work did; it felt like they were working on down-to-earth problems with heavy tools, while Hida seems to like studying those technical tools.
Anyway, you develop your interests in that first year... if you're still not 100% sure what you want to do, you go to a big department that's very active in many different fields... ;)
(Also, four number theorists isn't particularly small. The topology group consists of 2 people (we're getting another next year though!); the geometry group consists of 2 people; the combinatorics group consists of 1 person, I think there are five algebraists though they work on a variety of things...)
And once you split analysis into its subsets, the same sort of thing happens.
I think some schools are notable as being very strong in some field or another and pick up a large cloud of people. UCLA is more general, I think, with lots of very strong people - but no particular group really overwhelmingly big.
That's been my impression. Both Penn and UW Madison have very strong arithmetic geometry/number theory groups. But UCLA has a very good all around collection of excellent people.
(Which isn't to say its number theory isn't excellent, I just know less about it.)
Oh, that's what you mean. Yes, I did, but I also at that point didn't have any other options, and was only waiting on two - the visit convinced me that their decisions wouldn't change my mind.
(If you have multiple visit days, I really think you should go to both.)
Haha, fair enough. Oh, definitely, I'll be going to all of them, and I imagine I'll wait until the end, even if I do have a strong feeling one way or another.
I need to know if I'm doing this exercise correctly:
$f(z)$ has only one singularity in $z=0$, and it's a pole of order 7
$f(z)=-f(-z)$
$f(z)$ is analitic in $z=\infty$
$g(z)$ is analitic everywhere except of $z=\infty$, where it has a pole of order 7
I need to calculate $Res[f.g]$ in $z=0$
...
