Theorem 32.2 - every linear transformation has a corresponding matrix transformation

• The columns of a matrix represent what they do to the $n$-th basis vector.

• Creating a matrix like so: $[L({\vec{e}}_1)L({\vec{e}}_2)...L({\vec{e}}_n)]$ let’s us represent what $L$ does to each of the basis vectors, hence capturing what each of the column would to do the $n$-th basis vector.

• $[L]$ is the standard matrix of transformation $L$.

• So to find the standard matrix, we evaluate $L$ with ${\vec{e}}_i$ for each $n$.