Tagged Questions
4
votes
2answers
124 views
Proof that the $d$-th powers generate the $d$-th symmetric power of a vector space
Let $V$ be a $\mathbb{C}$-vector space of finite dimension. Denote its $d$-th symmetric power by $V^{\odot d}$. I am looking for a proof that $V^{\odot d}$ is generated by the elements $v^{\odot d}$ ...
2
votes
2answers
76 views
How can one see that $\operatorname{tr}(f\otimes g)=\operatorname{tr}f\operatorname{ tr }g$?
Suppose you have two free modules $M$ and $N$ of finite rank over a commutative ring $R$. Let's also take some $f\in\operatorname{End}_R(M)$ and $g\in\operatorname{End}_R(N)$, which gives a ...
11
votes
2answers
239 views
Why is it that $\det(\phi-x\text{id})=\sum_{i=0}^n (-1)^ic_ix^i$?
I'm trying to understand a certain formula for the determinant in a more general setting.
Say you have a free module $M$ of rank $n$ over a (commutative) ring $R$. Let ...
3
votes
0answers
74 views
Is $M\to M^{\vee\vee}$ injective when $M$ is free?
It's a common theorem that when $M$ is a finite-free $R$-module of rank $n$, there is a natural isomorphism $M\cong M^{\vee\vee}$, where $M^\vee$ denotes the dual. So $M^{\vee\vee}$ is also free of ...
