Recent content by aalma

Right, this gives ##g'(x)=f'(x)(f(a)-2f(x)+f(b))##. ##g'(t)=0## but how this says that ##f'(t)## must be 0? (How we use the assumption abou c?)

Yeah, nut then ##g(a)=0=g(b)## and since g is continuos on [a,b] and differentiable on (a,b) so by Roll's theorem there is ##t \in (a,b): g'(t)=0## but then how to use that ##g(c)<0##?

##(f(c) - f(a))((f)(b) - f(c)) <0## tells us that there are two cases: ##f(c) >f(a), f(b) ## ##f(c) <f(a), f(b) ##. I guess we need to define a new function here that let us use the Rolle's theorem.. But it is not clear enough how to do so.

Can you please take a look on the previous comment

Just to make sure, i think there is some confusion in the calculations in the example of ##S_3##? ##T_{\sigma}T_{\tau}## is a product of two matries and equals to the matrix ##T_{\sigma\circ tau}##, right? Now again though it is clear, what is the way to show it formally? Is what was already...

Thanks! In the first part, I am still confused of what ##a_{1,\sigma(1)}## and so on represent (are they the entries of ##I## after appling ##\sigma## and so the product of the ##a_{i, \sigma(i)}## is not zero just when ##\sigma=id##? In the other part, Can I say that ##T_{\sigma}...

Question: Let ##\sigma\in S_n## be a permutation and ##T_{\sigma}## be the matrix we obtain from ##I## by appling ##\sigma## on the raws of ##I## (I.e ##\sigma## acts on the rows of ##I##) . Then: 1. ##\det(T_{\sigma}) = sgn(\sigma) ## and 2. ##T_{\sigma} T_{\tau} =T_{\sigma\circ \tau}##, for...

Yes, I agree. But it is still hard for me to connect all things tohether. Can you provide some details of how this works (Starting with ##g\in H## how to find ##\lambda, v##? I tried to show that the given set is a subrep but I got stuck: Given ##g\in G, w\in V: g(w)=\lambda w## then why is...

Can you please explain how we use this (Schur's lemma, the version you mentioned) in this case. I do not understand how we start with this question reching this lemma

Above

Yeah, I guess that too, but what am I supposed to add then? They mention that "the fundamental group of ##SL(2,R)## is ##Z## and define by ##G## the universal covering group for ##SL(2,R)##", where do we exactly use this information? ##\rho:G\to GL(n,C)## so how it then gives rise to the rep...

Thanks! I have some questions if it is okay. *Does the "D" in ##D\pi## (for example) denote? *Can tou please mention the statement of the covering property you're using here. *Do we actually need to know here that ##sl_2(C)=sl_2(R)×C## (I am not sure if it something I can use or it is just like...

I cannot see how to connect this to my question

I am not quite sure of how this works, i.e. of what exactly I need to do with the hint. Any explanantion would be helpful!

Yes, that is true. In the way the question is given, I thought that I need to do the calculations for derivations too (which might differ, since the Leibniz rule is involved..).