Worksheet 19

1. Show that $V \cong W$ is an equivalence relation on vector spaces.

2. Is there a finite-dimensional vector space with $V \cong V \times V$? What about infinite-dimensional?

3. Let $\operatorname{Sym}_n(k)$ be the vector space of symmetric, recall $A = A^T$, $n \times n$ matrices. Let $\operatorname{AntiSym}_n(k)$ be the vector space of anti-symmetric, $A^T = -A$, $n \times n$ matrices. Is \(\operatorname{Sym}_n(k) \cong \operatorname{AntiSym}_n(k)?\)