Suppose that $A$ is an $m\times n$ matrix. Define the linear transformation $T$ by \begin{equation*} \ltdefn{T}{\complex{n}}{\complex{m}},\quad \lt{T}{\vect{x}}=A\vect{x} \end{equation*} Prove that the kernel of $T$ equals the null space of $A$, $\krn{T}=\nsp{A}$.