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 $h(\vec{\beta}_1)$, ..., $h(\vec{\beta}_n)$.