The prior section shows that a linear map is determined by its action on a basis. In fact, the equation
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 at all. We just need to find the 's to express with respect to the basis.
This section gives the scheme that computes, from the representation of a vector in the domain , the representation of that vector's image in the codomain , using the representations of , ..., .