Linear Algebra/Computing Linear Maps

The prior section shows that a linear map is determined by its action on a basis. In fact, the equation

$h(\overrightarrow{v})=h(c_1\cdot {\overrightarrow{\beta }}_1+\dots +c_n\cdot {\overrightarrow{\beta }}_n)=c_1\cdot h({\overrightarrow{\beta }}_1)+\dots +c_n\cdot h({\overrightarrow{\beta }}_n)$

shows that, if we know the value of the map on the vectors in a basis, then we can compute the value of the map on any vector $\overrightarrow{v}$ at all. We just need to find the $c$'s to express $\overrightarrow{v}$ with respect to the basis.

This section gives the scheme that computes, from the representation of a vector in the domain ${\mathrm{R}\mathrm{e}\mathrm{p}}_B(\overrightarrow{v})$, the representation of that vector's image in the codomain ${\mathrm{R}\mathrm{e}\mathrm{p}}_D(h(\overrightarrow{v}))$, using the representations of $h({\overrightarrow{\beta }}_1)$, ..., $h({\overrightarrow{\beta }}_n)$.