We suppose now that U and V are finite dimensional, and that , . Let be a libear mapping.
Choose bases: for U and for V.
Take a vector
in U; there exists a unique n-tuple of scalars
Therefore the linear mapping is totally determined by the data of the vectors .
We represent this data by the matrix:
If the coordinates of w.r.t. the given basis of U are and if the coordinates of w.r.t. the given basis of V are , they are related by the formulas:
We denote this in the following way:
A basis for U: p1(x)=1, [2(x)=x, p3(x)=x2, p4(x)=x3.
We have: , , , . Thus, the matrix of w.r.t the given basis is: .