We are interested in the form of the matrix associated to a linear operator after a basis' change.
Let be a n dimensional k-vector space over a field k. Let and be two basis of V. We shall call transition matrix from to the matrix associated to identity map of V in the two basis and namely
After the previous definition we have . Let .
so we have for all
or in matrix form
Now it is easy to find what relation is between associated matrix.
Let , two basis in ,
two basis in ,
. Then .
For those familiar with manifold, now or in the future, this is a very elementary example of local charts and maps and how they described a manifold morphism.
A vector space can be viewed as an abstract "manifold ". For a basis, the correspondence is a global map , linear transform is an application and is its expression in the local charts.
shows how the linear map looks like in another local system.