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Sep
30
awarded  Explainer
Aug
30
comment Alternative proof: Matrix $A$ is similar to $B$ iff $\lambda I - A$ is equivalent to $\lambda I - B$
I'm sorry, I've never heard this terminology. I assume it means $\exists P,Q\in GL(n,\mathbb K)$ such that $\forall \lambda \in \mathbb K$, $\lambda I - A=P^{-1}(\lambda I -B)Q$, right?
Aug
30
comment Alternative proof: Matrix $A$ is similar to $B$ iff $\lambda I - A$ is equivalent to $\lambda I - B$
I think that you should state the theorem more precisely: what values is $\lambda$ free to take? Since in this form it is obviously false: take $\lambda =0$.
Aug
30
comment Alternative proof: Matrix $A$ is similar to $B$ iff $\lambda I - A$ is equivalent to $\lambda I - B$
Matrix equivalence: en.wikipedia.org/wiki/Matrix_equivalence
Aug
29
comment The differential $\text d F_p$ is injective iff the pullback $F_p^*$ is surjective.
Thank you anyway; as for your last remark, I was indeed trying to prove this using the canonical isomorphism $T_pM \simeq (m_p/m_p ^2)^*$, where $m_p$ is the ideal of germs that vanish in $p$. The differential identifies with the dual of the mapping $\varphi _p$ induced on $m_{F(p)}/m_{F(p)}^2$ by $F_p ^*$. In this case the problem is to prove that $\varphi _p$ surjective implies $F_p ^*$ surjective.
Aug
29
comment The differential $\text d F_p$ is injective iff the pullback $F_p^*$ is surjective.
I didn't think about using the fact that $F$ is locally an immersion. Thank you!
Aug
29
accepted The differential $\text d F_p$ is injective iff the pullback $F_p^*$ is surjective.
Aug
29
revised The differential $\text d F_p$ is injective iff the pullback $F_p^*$ is surjective.
edited title
Aug
29
asked The differential $\text d F_p$ is injective iff the pullback $F_p^*$ is surjective.
Aug
27
revised Openness of path connected components of open subsets of $\mathbb C$
added 41 characters in body
Aug
27
comment Openness of path connected components of open subsets of $\mathbb C$
@LeeMosher I'm not sure I understand
Aug
27
answered Openness of path connected components of open subsets of $\mathbb C$
Aug
25
comment Sequence $\frac{(-2)^{n!}}{n^n}$ diverges
@Giiovanna there's a \leq that should be \geq
Aug
22
comment Why are germs of functions important?
I agree with @Marra that from your answer is not really clear why germs are useful. I was wondering about the same question of the OP since, from my limited perspective, the only difference made by the use of germs are those “it's easily checked that the definition doesn't depend on the representative” etc.etc. A concrete example may be useful.
Aug
21
comment Differential of the inversion of Lie group
Thank you @Siminore, indeed the second link was helpful
Aug
21
comment Differential of the inversion of Lie group
Hi Troy, thank you. Your proof seems to rely on the fact that: $$\frac {\text d}{\text {dt}}|_0(\sigma _1 \sigma _2)=\sigma _1 (0)\dot \sigma _2 (0)+\dot \sigma _1 (0) \sigma _2(0),$$ is that right? How do you prove that?
Aug
20
asked Differential of the inversion of Lie group
Jul
30
comment If $a>1$, prove that $\lim_{n \rightarrow \infty } a^n = \infty$
Remember the binomial theorem? $$(1+x)^n =\sum _{i=0} ^n \frac{n!}{(n-i)!i!}x^i$$
Jul
30
answered If $a>1$, prove that $\lim_{n \rightarrow \infty } a^n = \infty$
Jul
30
comment Algebraic proof of $\tan x>x$
@MarioCameiro if one can prove the equality beetween the power series definition of $\cos x$, $\sin x$, $\tan x$ and the segments in the picture, the visual proof can easily be made rigorous. As Barry said, in Rudin's book there's a similar approach to the trigonometric functions and the rigorous proof that I'm claiming is practically given. But, as you may expect, this one is ruled out by the “non calculus” requirement.