Let $ A: \mathbb{C}^2 \to \mathbb{C}^2$ be given by
$ A =\begin{bmatrix} 1 & 1 \\ -1 & -1 \\ \end{bmatrix} $
(with respect to the standard basis). Find a basis of $\mathbb{C}^2$ with respect to which the matrix reprsentation of $A$ is
$D = \begin{bmatrix} 0 & 1 \\ 0 & 0 \\ \end{bmatrix}$
So far, I have set up the problem as such:
$ D = C^{-1}AC$ and substituted it as
$\begin{bmatrix} 0 & 1 \\ 0 & 0 \\ \end{bmatrix} = C^{-1}\begin{bmatrix} 1 & 1 \\ -1 & -1 \\ \end{bmatrix}C$, in which $C$ is the basis. Thus far, I have only seen examples in which I was given the basis and asked to solve for the matrix representation. I am not sure what the problem means by "with respect to the standard basis" either. How do I go about solving this problem?