Proof that all 2x2 matrices with rational entries are similar to
t 0
0 t

or to

0 a
1 b
I know that similar matrices characterize the same linear map to different bases. So with both forms of matrices it should be possible to describe all linear maps from Q^2 to Q^2. But I have no idea how to show that.
Is it necessary to take a base of Q^2 and look what the two matrices do with it (or with the linear combinations of the base???). Do I have to find out what the base must be?