The transformation $R^2 -> R^2$ is represented by the matrix $\pmatrix{1&2\\1&0}$.
Find the eigenvalues and eigenvectors and decide the matrix representation of T with respect to a basis of eigenvectors.
So, I get that the eigenvalues are: $\lambda_1 = 2 $ and $ \lambda_2 = -1 $
This gives me the eigenvectors: $ \pmatrix{2\\1}$ and $ \pmatrix{-1\\1}$. However, I don't really grasp what to do when I want to represent the matrix transformation with respect to the eigenvectors. I've tried virtually everything I could think of but everything is wrong. What's the correct way to do this?
Note that the representation with respect to the basis of eigenvectors $v_1=\pmatrix{2\\1}$ and $v_2=\pmatrix{-1\\1}$ is the diagonal matrix with the eingenvalues on the diagonal, that is
$$\pmatrix{2&0\\0&-1}$$