Prove that there exists an ordered basis $\gamma$ for which $[T^*]_\gamma$ has a column of $0$s.

Question

$V$ is an $n$-dimensional vector space over a field $\mathbb{F}$. Assume that $T^*:V\rightarrow V$ is a linear operator on $V$ and $T^*$ is not an isomorphism. Prove that there exists an ordered basis $\gamma$ for $V$ such that $[T^*]_\gamma$ has a column of $0$s.

Answer 1

Since $T^*$ is not an isomorphism, there is some $V\ni a\not=0$ such that $T^*(a)=0$. Now try to find a basis containing $a$.