# All Questions

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### Wronskian is independent to the choice of a basis

How to show that the Wronskian is independent to the choice of a basis of $V$? Let $f_1(x), \ldots, f_n(x)$ be a basis of a vector space $V$. Then the Wronskian is the determinant of a matrix whose ...
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### Vertex Clique Cover and Chromatic Number

I have heard that the Vertex Clique Cover Number is equal to the Chromatic Number of the complement of a graph. But, I can't find a reference. Is this true? And, is it true for all graphs or just ...
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### Is this a consequence of independence?

Suppose the real random variables $X$ and $Y$ are independent; i.e. $$\mathbb{P}[\{X \leq x\} \cap \{Y \leq y\}] = \mathbb{P}[\{X \leq x\}] \cdot \mathbb{P}[\{Y \leq y\}].$$ Does it follow that ...
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### If $A\vec{x}=\vec{b}$ and $B\vec{x}=\vec{b}$ inconsistent, then $(A+B)\vec{x}=\vec{b}$ inconsistent?

The matrices $A$ and $B$ are of $n \times m$. Suppose there is a $\vec{b}$ such that both $A\vec{x}=B\vec{x}=\vec{b}$ have infinitely many solutions. Is it true that $(A+B)\vec{x}=\vec{b}$ also has ...
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### Is my use of Lagrange multipliers correct?

Given $n$ positive values. $x_i \ge 1 (1\le x_i \le n)$. Their sum is $k$. $$\sum_{i=1}^{n}x_i = k$$ Define the following value: $$\sum_{i=1}^{n}x_i(x_i-1)$$ Now use Lagrange multipliers to ...
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### Order of product of two elements in a group

Let $G$ be an abelian group. Let $a, b \in G$ and let their order be $m$ and $n$ be respectively. Is it always true that order of $ab$ is $lcm(m,n)$? What if $m$ and $n$ are coprime to each other?
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### Showing countability of $V:=\{f\in F^{\mathbb{N}}:\operatorname{supp}(f)$ is finite $\}$ and uncountability of $V'$

Let $F:={\mathbb{Z_2}}$ be the field with two elements and let $F^{\mathbb{N}}:=\{f:\mathbb{N}\to\mathbb{Z_2}:$ with usual pointwise addition and multiplication by scalers $\}$ Then define ...
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### What is Birkhoff's ergodic theorem for $GL_n$?

I was wondering about the general setup of a "noncommutative" ergodic theory. Instead of measurable maps into $\mathbb R$, we should consider measurable maps into $GL_n(\mathbb R)$, and instead of ...