Suppose that $\ltdefn{T}{U}{V}$ is a linear transformation, $B$ is a basis for $U$ of size $n$, and $C$ is a basis for $V$. Then the kernel of $T$ is isomorphic to the null space of $\matrixrep{T}{B}{C}$, \begin{equation*} \krn{T}\isomorphic\nsp{\matrixrep{T}{B}{C}} \end{equation*}