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Aug
13
comment How to come up with a counter example in linear algebra
The statement is generally false because the choice of $U$ isn't unique. You should note that the freedom of choosing $U$ is quite arbitrary. For example, consider the following matrix $\left[\begin{array}[rr]\ 1&0\\0&2\end{array}\right]$. $\langle e_1\rangle$ is an invariant subspace, so is $\langle e_2\rangle$. Well, how about choosing another complement of $\langle e_1\rangle$, say $\langle e_1+e_2\rangle$?
Aug
13
comment The uniqueness of a special maximal ideal factorization
@JohannesKloos if $M\ne0$, I doubt $MA=MB\implies A=B$ is still wrong, since the condition seems somewhat redundant if that's true. (I have tried to prove $MA=MB\implies A=B$. Once others pointed out the case of $M=0$, I immediately veered to believe that it's not generally true)
Aug
12
asked The uniqueness of a special maximal ideal factorization
Aug
10
comment Calculate a sum of elements over a finite field $\mathbb{Z}_n$
Please read this wiki
Aug
7
comment Geometrical interpretation of a group action of $SU_2$ on $\mathbb S^3$
@MarcvanLeeuwen I hope it's clearer now.
Aug
7
revised Geometrical interpretation of a group action of $SU_2$ on $\mathbb S^3$
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Aug
7
revised Geometrical interpretation of a group action of $SU_2$ on $\mathbb S^3$
added 489 characters in body
Aug
7
revised Integral inequality
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Aug
7
comment Does a Hermitian matrix remain Hermitian when transformed by an orthogonal matrix?
What do you mean by transformed by an orthogonal matrix?
Aug
7
answered Integral inequality
Aug
7
revised Geometrical interpretation of a group action of $SU_2$ on $\mathbb S^3$
added 76 characters in body
Aug
7
asked Geometrical interpretation of a group action of $SU_2$ on $\mathbb S^3$
Aug
6
comment How to prove : $\int\limits_{1}^{+\infty} x.f(x)dx$ is convergent
@EricAuld As far as I know, there're some Soviet textbooks containing it. The proof isn't hard, however, by the second mean value theorem for integration, and Cauchy's criterion.
Aug
5
comment How to prove : $\int\limits_{1}^{+\infty} x.f(x)dx$ is convergent
Another version: replace condition 2 by that $\phi$ is monotone but not necessarily differentiable.
Aug
5
accepted An easy half of Quillen-Suslin theorem
Aug
5
comment An easy half of Quillen-Suslin theorem
Your reasoning shows that $\operatorname{rank}A(v)\ge n-k$. However, since $BA=0$, we have $B(v)A(v)=0$, therefore $\operatorname{rank}A(v)\le n-k$, Q.E.D.
Aug
3
awarded  Enlightened
Aug
3
awarded  Nice Answer
Aug
3
comment prove that there is $c \in (a,b)$ (about calculus)
@CrMT In fact, the construction of $g$ is essentially related to the theory of ordinary differential equations. It's a basic trick called integrating factor. You can search it online. Anyway, learning something deeper is seldom harmful.
Aug
3
comment An easy half of Quillen-Suslin theorem
Is it the application of the short five lemma on the following diagram: $\require{AMScd}\begin{CD}R^n@>>>R^{n+l}@>>>R^l@>>>0\\@VVV@VVV@VVV\\R^n@>{A}>>R^m @>>>V @>>>0\end{CD}$