A basic question on diagonalizability How to prove that the following linear map :
$T : \Bbb R^2 \to \Bbb R^2$ defined by $$T(e_1)=e_2$$
$$T(e_2)=0$$ 
can't be diagonalized with respect to any basis. Here $e_i$s are standard basis.

A transformation is diagonalizable iff its eigenvectors form a basis of the vector space.  In this case, $T$'s only eigenvectors are the multiples of $e_2$.  Since these do not form a basis of $\mathbb R^2$, $T$ is not diagonalizable.
