Homework 10 - Bases and isomorphisms
Show that a $n \times n$ matrix is a change of basis matrix if and only if it is invertible.
We define a relation $\sim$ on $\operatorname{Mat}_{m,n}(k)$ by writing $A \sim B$ if there exist an invertible $m \times m$ matrix $C$ and an invertible $n \times n$ matrix $D$ with \(B = CAD\) Show that $\sim$ is an equivalence relation. From the previous problem, $A \sim B$ is equivalent to saying that $A$ has $B$ as its matrix representation in different bases.
Show that if $V_1 \cong V_2$ and $W_1 \cong W_2$, then $V_1 \times W_1 \cong V_2 \times W_2$.
Suppose $V$ is a finite dimensional $k$-vector space with a positive definite inner product $\langle -,- \rangle$ and $U \subseteq V$ is a subspace. Show that the composition \(U^{\perp} \to V \to V/U\) is an isomorphism $U^{\perp} \cong V/U$. (Hint: compute the dimensions of each.)
The dual space of a vector space $V$ is the space of linear transformations $V \to k$, called $\operatorname{Map}(V,k)$ previously. The dual space is usually denoted by $V^\ast$. Show that if $V \cong W$, then $W^\ast \cong V^\ast$. (Hint: given an isomorphism $T: V \to W$ and linear transformation $\psi: W \to k$ how can you combine them to get a linear transformation $V \to k$?)