then if you set the matrix as unknown $$ \begin{bmatrix} a & b \\ c & d \end{bmatrix} \cdot \begin{bmatrix} 2 \\ 5 \end{bmatrix}=\begin{bmatrix} 24 \\ 16 \end{bmatrix} $$
@Sha Galois theory is quite fundamental in my opinion 'cuz it tells you a proof of the fact that quintics cannot be solved in terms of radicals (I suppose you already know this story)
galois theory can be applied to classical geometry. You can show that it's impossible to trisect an angle, double a cube or square a circle (for the last one you'll need the transcendence of $\pi$)
i have a question : For a point x, let $N_{x}$ denote the collection of all nbd's of $x$. Define $w_{f}(x) = inf_{V \in N_{x}}\{sup_{y,z \in V} |f(z) - f(y)|\}$. Then $f$ is cts. at $x$ iff $w_{f}(x) = 0$. Can someone explain what is happening here?
I find myself in sometimes in these situation where i try to seek a good material for some specific problem related to mathematics and usually this can be very time consuming. Usually happens when there is no-one to ask for help.
Then if you had that someone to ask it from the situation would be different.
You find various books as good material or what is your number one type of material you seek knowledge from @BalarkaSen ?
@ÍgjøgnumMeg Miete ist im Allgemeinen nicht so günstig, obwohl ich Glück hatte mit der Wohnung. Es gibt auch Studentenwohnheime, bei denen man sich bewerben kann
Heidelberg ist aber günstiger als München, Frankfurt, Berlin, Hamburg ...
@MatheinBoulomenos kommt mir auch so vor, ich werd mich auch in Augsburg bewerben aber es kommt mir so vor als wär miete in bayern generell teurer als in anderen bundesländern
I once showed that a polynomial of degree 5 where the highest coefficient was $68$ or something like that is irreducible and my argument involved the well-known fact that $1371294863$ is a prime number
Though I think he kinda felt bad since the average was like, 40%. He changed the next test, probably wanting to make it shorter, and it was more interesting but the average stayed the same. Final was a bit better and it was also legitimately fun
I don't know if they are a good measurement of understanding, probably homework where you don't have time restrictions and have access to your notes and the exercises can be more difficult are a better measurement of that
But I think that oral exams are the best measurement of understanding
Profs can usually tell really well if someone understands the subject if they just talk a little bit about it
But yeah he's getting better so there's hope this test won't be stupid, it helped that the material of the final was a lot of Sylow combo and honestly I could major in Sylow
Pretty well, I did the functional analysis exam and it went well, I still have to do a foundations of math/logic and a commutative algebra one before the new semester begins, but those are topics I like so it's fine
I took basic course in linear algebra and i think i understood most of the concept but then when you had to inverse 5x5 matrices in exam that didn't go exactly well. Also no calculators were allowed
also it included various decompositions for example lu decomposition, svd
@AlessandroCodenotti what did you cover in commutative algebra? My course was combined with non-commutative, homological algebra and category theory, so my commutative algebra knowledge is mostly self-taught
I have two primes $p, q$ such that $p^2 \equiv 1 \pmod{q}$ and $q^2 \equiv -1 \pmod{p}$. I think only $(p, q) = (3, 5)$ is possible. How would I approach this?
Ideally suggest something rather than full disclosure.
@MatheinBoulomenos Standard stuff, modules, Hilbert's basis theorem, the nullstellensatz, primary decomposition, localisations, going up and going down theorem, we folloed Miles Reid book
How does that bit work?
How is
$$\cos(\arcsin(x)) = \sin(\arccos(x)) = \sqrt{1 - x^2}$$
