Let \(S\) and \(T\) be linear transformations \(V \rightarrow W,\) where
\(\operatorname{dim} V=n\) and \(\operatorname{dim} W=m\)
a. Show that \(\operatorname{ker} S=\operatorname{ker} T\) if and only if \(T=R
S\) for some isomorphism \(R: W \rightarrow W\). [Hint: Let
\(\left\\{\mathbf{e}_{1}, \ldots, \mathbf{e}_{r}, \ldots,
\mathbf{e}_{n}\right\\}\) be a basis of \(V\) such that
\(\left\\{\mathbf{e}_{r+1}, \ldots, \mathbf{e}_{n}\right\\}\) is a basis of
\(\operatorname{ker} S=\operatorname{ker} T\). Use
Theorem 7.2 .5 to extend \(\left\\{S\left(\mathbf{e}_{1}\right), \ldots,
S\left(\mathbf{e}_{r}\right)\right\\}\) and
\(\left\\{T\left(\mathbf{e}_{1}\right), \ldots,
T\left(\mathbf{e}_{r}\right)\right\\}\) to bases of \(\left.W .\right]\)
b. Show that \(\operatorname{im} S=\operatorname{im} T\) if and only if \(T=S R\)
for some isomorphism \(R: V \rightarrow V\). [Hint: Show that
\(\operatorname{dim}(\operatorname{ker} S)=\operatorname{dim}(\) ker \(T)\) and
choose bases \(\left\\{\mathbf{e}_{1}, \ldots, \mathbf{e}_{r}, \ldots,
\mathbf{e}_{n}\right\\}\) and \(\left\\{\mathbf{f}_{1}, \ldots, \mathbf{f}_{r},
\ldots, \mathbf{f}_{n}\right\\}\) of \(V\)
where \(\left\\{\mathbf{e}_{r+1}, \ldots, \mathbf{e}_{n}\right\\}\) and
\(\left\\{\mathbf{f}_{r+1}, \ldots, \mathbf{f}_{n}\right\\}\) are bases
of ker \(S\) and ker \(T,\) respectively. If \(1 \leq i \leq r,\) show that
\(S\left(\mathbf{e}_{i}\right)=T\left(\mathbf{g}_{i}\right)\) for some
\(\mathbf{g}_{i}\) in \(V,\) and prove that \(\left\\{\mathbf{g}_{1}, \ldots,
\mathbf{g}_{r}, \mathbf{f}_{r+1}, \ldots, \mathbf{f}_{n}\right\\}\) is a basis
of \(\left.V .\right]\)
