How do we find the eigenvectors of a linear transformation? How do we find pleasing (or computationally simple) matrix representations of linear transformations? Theorem EER and Theorem SCB applied in the context of Theorem DC can answer both questions. Here is an example.
Now we compute the eigenvalues and eigenvectors of M1. Since M1 is diagonalizable, we can find a basis of eigenvectors for use as the basis for a new representation.
The eigenvectors that are the basis elements in B are the eigenvectors of the linear transformation, relative to the standard basis. For different representations the eigenvectors take different forms, relative to other bases. What are the eigenvectors of the matrix representation M2?
Notice that the eigenvalues of the linear transformation are totally independent of the representation. So in a sense, they are an inherent property of the linear transformation.
You should be able to use these techniques with linear transformations on abstract vector spaces — just use a mental linear transformation transforming the abstract vector space back-and-forth between a vector space of column vectors of the right size.