## Matrix Representation of Linear Operators

Let us consider a k-vector space morphism of vector space between two finite dimensional k-vector spaces.over a field k. We shall call such an object linear transform or linear operator.
So let be two k-vector and the set of k-linear operators from to . If then we have

Let be a basis in and be a basis in . For all there are such as .
Now

But if we write we have

and , as form a basis in we have for all i from 1 to m

Let be We shall call the matrix representation of in the two basis and write
Let vectors in written in column form.
then relation

has an analog

deduced by (1)