@LeakyNun "What truly is logic? Who decides reason? My questions take me through the physical, to the metaphysical, to the delusional, and back. And I have made the most important discovery of my career, the most important discovery of my life. It is only in the mysterious equations of love that one finds any logical reasons. You are the reason I am here today. You are all my reasons."

@NV-US Nope... $\begin{bmatrix}2&1\\0&1\end{bmatrix}+\begin{bmatrix}1&0\\0&2\end{bmatrix}$

watch from 5:15 @KasmirKhaan

@Jasper no, not in that sense

in the sense that they write hand-wavy proofs and don't see where the hand-waviness is

Q: "What is love?" A: "Baby, don't hurt me. No more."

@robjohn ty

@KasmirKhaan how can I help you?

well let see first if the first implication is correct

assume f is injective

@LeakyNun That's not an argument...

f(x) = f(y) , x=y

@robjohn ?

since f(e_G) = e_H

@LeakyNun sorry, a Monty Python reference

ok

then it is unique

so the ker(f)= {e}

ok

no two elements maps to the same

so thats it

for ==>

for the other direction

ker(f) = {e} ==> injecitve

assume that f(x) = f(y) with x different from y

f(x) f(y) = f(xy) by hom

since we assume f(x) = f(y)

f(xy) = f(xx)

this means xy=xx

or y=x

contradicting the fact that x and y were not equal

is that good or bad proof?

why does f(xy) = f(xx) mean xy=xx?

because f(xy) = f(x)f(y)

f(xx) =f(x) f(x)

f(x) f(y) = f(x) f(x)

these elements have inverse because the image is a subgroup

multipliying by f(x)^-1

or wait

@LeakyNun yeah thats it

what do you get?

f(x) = f(y)

so how do you get x=y?

we get a contradicting

if x is different from y

@LeakyNun Argument Clinic

and f(x) = f(y)

we multiply by f(x)^-1

I don't see where the contradiction is.

we get f(y) f(x)^-1 is in the kernel

but we know only {e} is in the kernel

that is a contradiction

ok

@robjohn "Argument clinic" can be the name of a complex analysis tutorial, lol.

@Jasper $e^{ix}$ will have them spinning

well is there some lectures on group actions?

i really want to get this topic but from what i read its not that clear

@KasmirKhaan Are you still having a hard time understanding group actions?

@KasmirKhaan have you looked at the Wikipedia page?

@Jasper indeed

@robjohn i did but its cryptic to me

I mean I need like something to get my intuition going

@KasmirKhaan Do you at least understand the definition?

i do

we have 1.a=a for all a in the set A

and (gh)a = g (ha)

gh is done in G

Maybe look at a few examples.

hmm i have one in perticular that i did could not make sense of

Sometimes, there isn't too much intuition meant for something. You just need to accept it, lol.

it was the question proing GL_2( Z/2) is isomorphic to S_3

anon told me i could map

a= (1 0 )

b ( 0 1 )

and c ( 1 1 )

those are vectors

he said this was a group action , but i dont see how

@KasmirKhaan Did you try writing down what it means for that to be a group action?

@robjohn HAHAHAHAHA!

This is abuse.

@TobiasKildetoft hmm it is a permutation on a set of elements

oh wait ._.

hmm

Those words do not go together in that order

ga permutes A

no

like (123) [1] = [2]

hmm

then what is it am missing

$ga$ is an element of $A$

$g$ permutes $A$

okay so the fact that multiplying by g permutes A

that is the definiton of an action ?

hi @TobiasKildetoft

Do you see how the matrices act on those vectors?

@LeakyNun Hi

how would you suggest one to improve on logic?

@LeakyNun You mean formal logic, or just logical arguments?

@TobiasKildetoft latter

Hey there people!

hey dami

@LeakyNun Just practice. Learn the very basic rules of logic, and then practice

@Daminark Hi

@TobiasKildetoft what are the basic rules of logic?

@TobiasKildetoft I dont see how g acts

@KasmirKhaan have you heard of syllogism?

@KasmirKhaan So you have a matrix $X$ and a vector $v$ and you want to know what $Xv$ is?

How's it going?

@LeakyNun I dont know where you come up with these words =p answer is no

@LeakyNun I just mean the basic rules of the logical connectives, such as "and" and "implies"

@KasmirKhaan it's where you get "Q" from "P" and "P implies Q"

hmm

I need to take that course =p

@LeakyNun Also, learn the rules without ever learning the latin names for them. Those names are only a hindrance

@TobiasKildetoft ugh, those Latin names are only for quoting

I don't know how I would quote otherwise

Salut, @Ted

Hi @Tobias

resalut, Leaky

Hi @Ted

@LeakyNun Don't quote. The rules are simple enough that you don't need to name them to use them

@TedShifrin Hi

Hi @Alessandro

@TobiasKildetoft I'm trying to introduce them to Kasmir

@TedShifrin wie sagst du "was ist X" in Frankosisch?

@TobiasKildetoft How does matrices act on a vector?

Qu'est-ce que c'est que X?

@KasmirKhaan Well, how do you usually multiply matrices with vectors?

(probably the most verbose phrase for something simple)

@TedShifrin right

Französisch?

@TobiasKildetoft well using row dot column

I feel sad for those who were never taught the meaning of matrices.

@TedShifrin right

@KasmirKhaan Then use that

@TobiasKildetoft the matrix is what in this case?

@KasmirKhaan The actual matrices you had.

@TedShifrin how do you justify finding eigenvector by repeatedly applying the matrix on a vector?

@TobiasKildetoft 3by 2?

@TobiasKildetoft I just finished 9.3

so slow

Anyone got a hint on how to show that 0 is the only critical point of a homogeneous polynomial? It came up in a homework problem and I'm at a bit of a loss. Basically the only think I know about homogeneous polynomials is the Euler formula.

@KasmirKhaan No, where would you get those from? You had $2\times 2$ matrices

That's called the power method. You only find the eigenvector corresponding to the dominant eigenvalue.

@Kevin: That's all you need.

@TobiasKildetoft I had a = ( 1 0 ) b = ( 0 1) and c = ( 1 1 )

should all the vectors be there?

@LeakyNun 9.3 is an important result that comes up a lot in group theory

@Kevin You mean $0$ is the only critical value, don't you?

@KasmirKhaan Those are the vectors. You also has matrices

@TobiasKildetoft isn't 9.4 more important?

Hey @Ted!

Hi Demonark

@TedShifrin Okay I was wondering about that. Showing that 0 is the only critical value is easy peasy. But the homework says critical point

Maybe thats a typo

@LeakyNun 9.3 is used more often, as it is a very nice and general statement

In G&P it's value.

The usual application is to show that $f^{-1}(c)$ is a manifold for any $c\ne 0$.

It is called the normalizer/centralizer theorem

@TobiasKildetoft How do you prove $\forall x \varphi(x) \vdash \neg \exists x \neg \varphi(x)$?

Ah, this problem may be from G&P I'm not sure. Its not cited to anything

btw I don't quite have any idea on 9.4 @TobiasKildetoft

@LeakyNun I don't :)

@TobiasKildetoft hmm so we have a matrix acting on those vectors a ,b ,c

@LeakyNun Well, you should use the previous parts of 9

@TobiasKildetoft what should the matrix be ?

@KasmirKhaan Your group literally consists of matrices

And actually for the problem I only need that 0 is the only critical value. But for every homogenous polynomial I cook up, 0 is also the only critical point and that's what the hint in the homework says to prove, so I figured it was worth thinking about in case it comes up again

@TobiasKildetoft oh!! lol so stupid

@TobiasKildetoft ahh you mean those matrices okay ><

@Kevin: Well, let's think about it.

let me try it now :D

9.3 what are you talking about guys ? =p

So the gradient has as its components a bunch of homogeneous polynomials of degree $k-1$.

I wa sjust about to say that

We're fine as long as $f(x)\ne 0$. But suppose $f(x)=0$.

@KasmirKhaan It is a set of exercises I made and that Leaky is working through

hmm, apparently you say p-Sylow subgroup

@TobiasKildetoft oh nice :D

while I say Sylow p-subgroup

@Leaky: I'm with you.

@Ted choose your side :P

yay :D

The $p$ goes with the subgroup, not with Sylow's name.

@LeakyNun We say Sylow $p$-subgroup in Danish, but I recall seeing it the other way more often in English

Unless the subject was acutlly investigated by Sylow's evil tiwn, p-Sylow

It makes it possible to shorten to $p$-Sylow

@Kevin: It's totally false.

You can give an immediate counterexample (but take degree > 2).

Okay, I'll try to think of the counter-example

I knew it was false for fancy reasons: In algebraic geometry we have plenty of hypersurfaces that have singular points.

@TobiasKildetoft so from what i understood , we take a matrix A , we do Aa , Ab and Ac and we see what the matrix do to those vectors right?

@KasmirKhaan right

:D

Ya I knew there was some connection here because I only vaguely know that some areas of higher mathematics, coinciding zeroes of homogeneous polynomials are veyr important

first row ( 1 0 ) second row ( 1 1 )

So I figured there was some way to 'nuke the mosquito' as I like to say

acting on ( 1 1 )

= ( 1 0 )

@Tobias: Apparently Lang says p-Sylow. Ugh.

But he talks about p-subgroups. Stooopid.

sylow theorem is what we gonna be doing soon on class

also galois

@TedShifrin As long as people are consistent when Sylow is replaced by Hall

(I can't guarantee that I am)

Does anyone know which software this is?

(cc @user21820)

LOL ... I'm out of this argument ... My only connection to algebra is having written a little book :P

@TedShifrin an awesome book I might add

Which brings me to another question: did I coin the phrase 'nuke a mosquito' to mean to sue a very general, powerful theorem to answer a relatively trivial question, or did i hear that from someone and now cant remember?

You haven't even read it, Kasmir, so shaddup.

@Kevin: I've heard comparable phrases if not that particular one.

@TedShifrin I did :D not all ofc, but I need to follow the book the teacher gave us, because if Iread it full now, i wont be in time to follow up

you did group actions with geometry

and many other things

Well, so many beautiful examples come from geometric objects — not to mention important applications. (Google "homogeneous spaces" and "symmetric spaces".)

but I still did not have a serious read of it =p i will as soon as we get into rings

promice :D

hmm okay

@TobiasKildetoft how do I know that phi_g must be identity?

Still waiting on Category Theory: A Geometric Approach

@Daminark Geometry: A Category Theoretical Approach

But yeah so are we talking group actions here? What's going on?

where we have categories of polygons

@TedShifrin whats lie group

@KasmirKhaan a group that lies

@LeakyNun Where do you need that?

@LeakyNun really ?:o

@TobiasKildetoft to prove that every element in the normalizer is also in the centralizer

@KasmirKhaan nah, Lie was a dude

@LeakyNun how dare you -.-

@LeakyNun Use 9.3 and consider orders

i thought it was adefiniton like faithfull action

@Kasmir: It's a group that has the structure of a manifold (smooth surface in some dimension), and for which group operations are smooth functions.

@TedShifrin well this is way beyond my understanding so far =p ill keep reading the book and doing some exercices untill then:D

thanks yall for help :)

@Kasmir: any matrix group will do. :)

How does one solve equations of the form $f(x) = xg(f(x))$ for $f(x)$?

@Axoren for example?

I've come across that specific expression in an algebraic exploration.

I don't have a specific $g$ in mind.

@TedShifrin AhHa! Take $f(x,y) = x^3 + y^3 - x^2 y - y x^2$, then the entire line $y=x$ is critical. Oh, interesting, its one big saddle of the graph in $\mathbb{R}^3$

No idea.

@Kevin: Easier. What if $f(x,y)=xy^2$?

If only, @TedShifrin

Huh?

Oh whoops. That wasn't directed at me.

My name's Kevin

LOL, I don't think I knew that.

My mistake.

You keep asking innocent but impossible questions :)

This one just got posted on main and it's surprisingly annoying. I don't have the answer yet. Is there a path from $(0,0)$ to $(1,1)$ along which the force field $\vec F = (xy^2,y)$ does zero work?

Oh, I just figured it out.

The questions I ask are things that appear as roadblocks in my attempts to solve practical problems.

Let $A \in M_2(\Bbb R)$ such that $A^3=A$ and $A \ne 0,I,-I$. Find $\omega \in \Bbb R$ such that $\omega I - A$ is invertible.

The problem is that the solutions I want are the ones that solve the problem perfectly.

@Ted any hint?

@Leaky: I'm working on the multivariable calc problem and writing it up.

ok