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 ( \overset{→}{v} ) = h ( {c}_{1} ⋅ {\overset{→}{β}}_{1} + … + {c}_{n} ⋅ {\overset{→}{β}}_{n} ) = {c}_{1} ⋅ h ( {\overset{→}{β}}_{1} ) + … + {c}_{n} ⋅ h ( {\overset{→}{β}}_{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
\overset{→}{v} at all. We just need to find the
c's to express \overset{→}{v} with respect to the
basis.

This section gives the scheme that computes, from the representation
of a vector in the domain {R e p}_{B} ( \overset{→}{v} ), the
representation of that vector's image in the codomain {R e p}_{D} ( h ( \overset{→}{v} ) ), using the representations of h ( {\overset{→}{β}}_{1} ), ..., h ( {\overset{→}{β}}_{n} ).

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