$\text{Hey I have a general question about linear transformations as long as I confirm that } L(v_1+v_2) = L(v_1)+L(v_2), \\ \text{ and } L(\alpha v) = \alpha L(v) \\ \text{do I need to show}$ $L(\alpha v_1 + \beta v_1) = L(\alpha v_1) + L(\beta v_2) \text{Or is this already inferred}?$
Lol I'm taking Galois theory at the moment and while as a class it's perhaps somewhat less fun than combinatorics, I feel somehow like more than anything I've done before, this resonates with me
It's very interesting for sure. I didn't quite absorb stuff as well in this class since I kinda jumped into a class where I didn't know much GMT, Fourier, or PDE
And there were a few cases where the stuff just happened. Less PDE, really just one person's talk involved it, but at the beginning, Fourier and not knowing the basics of GMT made wrestling with the class trickier
Since it didn't teach the subject from the ground up, really people just started presenting papers published in the last few years
Hmm, I feel it's a bit early to say with much detail which direction I'm gonna be going but of the things I've seen so far, I feel like Galois theory and combinatorics are the two subjects where I enjoy solving problems the most
And if I had to guess, Galois theory is where I'm gonna have more luck. I've had some solutions in combo that made me proud of myself but I dunno if I would be able to have the consistent creativity of the type you need to really do that subject as my main thing
Also I know there's another theoretical compsci professor who is very combo-ish, though it seems that guy's work feels closer to math that's kinda sophisticated as opposed to clever counting arguments (mathematical logic, Fourier analysis, etc)
Yeah true algebraic combo sounds fun. It's all stuff I'd like to eventually explore
Oh so funny thing, I just realized that there have been so many incidents where I would find the inverse of an element in some field extension in a way more complicated than usual manner. These things often came up as, find the inverse of the root of (something), and that's so easy to do
@Ted yeah I did see the recommendations, thank you very much!
$p(x) = a_nx^n + \ldots + a_0$, and since $p(\alpha) = 0$ you can just write $1 = \alpha(-\frac{1}{a_0}(a_n\alpha^{n-1} + \ldots + a_1)$
Yeah you only need the first column of the inverse, which in small cases that I've usually had to deal with by hand (e.g. 3x3) has often been reasonably quick
Lol maybe you were thinking of $\mathbb{Z}[x]$ or something? Though usually it goes the opposite way if anything, you try to assume it's a PID and then... erm.. rip
alright, so: let $G=N\rtimes H$ be the semi-direct product. My book says that each $g\in G$ can be written uniquely as $(n,e)\cdot(e,h)$. However, why should this be unique? Can’t we just have $g=nh=n’h’$?
if $N \cap H = \lbrace e \rbrace$ then $nh = n^\prime h^\prime$ gives $(n^\prime)^{-1}n = h^\prime h^{-1}$. The left-hand side is where? And the right-hand side is where?
Okay, so we have $nh=n'h'$. We can then write $n'^{-1}n=h'h^{-1}$, and since $N\cap H=\{e\}$, we know that $n'^{-1}n=h'h^{-1}=e$, and hence $n'=n$ and $h'=h$
