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

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

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

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