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

${\displaystyle h({\vec {v}})=h(c_{1}\cdot {\vec {\beta }}_{1}+\dots +c_{n}\cdot {\vec {\beta }}_{n})=c_{1}\cdot h({\vec {\beta }}_{1})+\dots +c_{n}\cdot h({\vec {\beta }}_{n})}$

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 ${\displaystyle {\vec {v}}}$ at all. We just need to find the ${\displaystyle c}$'s to express ${\displaystyle {\vec {v}}}$ with respect to the basis.

This section gives the scheme that computes, from the representation of a vector in the domain ${\displaystyle {\rm {Rep}}_{B}({\vec {v}})}$, the representation of that vector's image in the codomain ${\displaystyle {\rm {Rep}}_{D}(h({\vec {v}}))}$, using the representations of ${\displaystyle h({\vec {\beta }}_{1})}$, ..., ${\displaystyle h({\vec {\beta }}_{n})}$.