In M340L or M341 you learned about the null space of a matrix, and about the column space. We learned how to find bases for these spaces in Appendix A. Now we're going to develop the analogous structures for linear transformations.

If $L: V \to W$ is a linear transformation, then the kernel of $L$ is the set of all vectors that are killed by $L$. That is: $$Ker(L) = \{ {\bf v} \in V | L({\bf v})=0\}.$$ This is a subspace of $V$, and is closely related to the null space of the matrix $[L]_{\mathcal{DB}}$. If you take the coordinates (in the $\mathcal{B}$ basis) of the elements of $Ker(L)$, you get exactly $Null([L]_{\mathcal{DB}}$.

The dimensions of these spaces are related to the rank of $L$ (or equivalently, the rank of $[L]_{\mathcal{DB}}$). The dimension of the range is the rank of $L$, and the dimension of the kernel is the dimension of $V$ minus the rank of $L$. In other words: $$dim(Range(L)) + dim(Ker(L)) = dim(V).$$ This is sometimes called the dimension theorem.