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
such that
.
Then:

Therefore the linear mapping is totally determined by the data of the vectors .

Denote: .

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*: *p*_{1}(*x*)=1, [_{2}(*x*)=*x*,
*p*_{3}(*x*)=*x*^{2},
*p*_{4}(*x*)=*x*^{3}.

.

We have: , , , . Thus, the matrix of w.r.t the given basis is: .