user153330

@TedShifrin thanks, yes i forgot that fact but i figured it out after :)

Hi! I have $A\in\mathcal{M}_{3\times 3}(\mathbb{R})$ and $A^\dagger A=k\operatorname{Id}$ where $k\in\mathbb{R}_{>0}$, also we assume that $A$ has an eigenvalue $\lambda$ with multiplicity $3$ (denote the eigenvectors by $a_1,a_2,a_3$. Consider the following operator: $M=k(A^t-A)$ I showed that $Ma_1=k(\frac{k}{\lambda}a_1-\lambda a_1)$, now I suspect that $\lambda^2=k$ but I can't show it from $A^\dagger A=k\operatorname{Id}$ alone, can anyone shine light here with some hint?

i will do my best as a youngster to take this matter seriously

and thanks for giving me some of your experience on this topic

@TedShifrin sorry to hear that

@TedShifrin sorry i didn't notice ur first comment

@TedShifrin i'm especially interested in your case since you're the oldest among us :) do you ever experience back pains?

how do you guys survive working without getting back pain?

thanks a ton @TedShifrin

@TedShifrin ohh that's true :)

@TedShifrin lol can't think of anything easier unfortunately :P

$e_2,e_3,...$ ?

@TedShifrin easier than your $e_1,e_3,e_5,\ldots$ : ) ?

@TedShifrin essentially yeah

it's not a basis

$e_1=(1/\sqrt{2},1/\sqrt{2},0,0,...), e_2=(0,0,1/\sqrt{2},1/\sqrt{2},...)$ would this construction work?

@TedShifrin i can't think of any other simple basis for $\ell_2$ unfortunately

@TedShifrin yeah standard basis but e_1 was different

@TedShifrin like it just goes on $e_3=(0,0,1,0,...)$ but this is useless as you showed

@TedShifrin oh that's true

closed implies complete, that's why i need it to be non-closed

@TedShifrin an orthonormal system A is closed in V iff $\sum_{n=1}^\infty |\langle v,e_n\rangle|^2=\|v\|^2$ for all $v\in V$

@TedShifrin i tried making a non-closed orthonormal system $e_1=(1/\sqrt{2},1/\sqrt{2},0,...), e_2=(0,1,0,0,...), ...$

i would appreciate any hint on how to find it

hey guys! what's an example of an incomplete orthonormal system in $\ell_2$?

@TedShifrin i think i got what you meant, i will work it out and answer you

@TedShifrin Hi Ted! wanna take a look at my quick confirmation question here chat.stackexchange.com/transcript/message/56059427#56059427 ?

hey everyone! i have a bit of a trouble with a question, so I'm tasked with finding the critical points of a map (meaning the points for which the Jacobian doesn't have full rank) this function $f:(x,y,z)\mapsto(xy,z)$ (and it's easy to show that $x=y=0$ is the only way a critical point may arrive) but what I have doubts about is that I should find the critical points of $f|_{S^2}$, are they simply $x=y=0$ with $|z|=1$?

@epic_math u can find the answer here math.stackexchange.com/questions/1131845/… (deleted my question since it was duplicate)

lemme ask on math.stackexchange

@epic_math $$o(H_1\cap … \cap H_n)=\dfrac{o(H_1)\cdots o(H_n)}{o(H_1\cdots H_n)}$$

@epic_math yes

@epic_math thx

can anyone propose a hint here?

and i have to show that they all have 2 as an index

i'm struggling a bit with this question, i have 3 non-trivial subgroups of a group such that their union is the group itself

any knowledgeable person in abstract algebra here?

hey guys

@Everstudent by the way if u can solve that problem please reach out to me i'm also in need for such help

@Everstudent i try to just write the essential definitions theorems and ideas of proofs, so it doesn't take much mental energy when writing

@Everstudent i have that same problem, and it holds me back, i still don't know how to overcome it

@Everstudent motivation is a bitch, persistence is key

you can't sit there and expect to happen on a perfect book that has all the motivations and subtleties in it

@Charlie yes, but what does mean in terms of the definition?

hey folks! i have a question concerning the torsion, my book defines it as ${T^b}_{ac}={\Gamma^b}_{ac}-{\Gamma^b}_{ca}$ but then it uses this notation ${{T_a}^c}_b$ and I have no clue of what it means, can anyone shine some light on this?

@WeiZhong seems like it crashed again

@MartinSleziak sorry that was my mistake