Let $\mathscr{V}$ and $\mathscr{W}$ be finite-dimensional vector spaces. The nullspace of a linear transformation $L: \mathscr{V} \rightarrow \mathscr{W}$ is the set of $X \in \mathscr{V}$ such that $L(X)=0$. It was shown in Exercise 3.23 on page 162 that the nullspace is a subspace of $\mathscr{V}$.
(a) Let ordered bases $B$ and $\overline{B}$ for $\mathscr{V}$ and $\mathscr{W}$ be given and let $M$ be the matrix of $L$ with respect to these bases. Show that $L(X)=0$ if and only if $M T_B^C X=0$.
(b) Show that $T_B^C$ restricts to an isomorphism of the nullspace of $L$ with the nullspace of $M$.
I understand how to do part (a), but I don't understand what part (b) is asking. What exactly does it mean for a matrix to restrict to an isomorphism. I have a feeling that this has a connection with part (a).
$T_B^C$ is the coordinate matrix for the ordered basis $B$.