Finding the standard matrix of the transformation, is it unique?

Question

I have a question that asks:

For the linear transformation given, find the standard matrix of the transformation:

$T: R^2 \rightarrow R^2 $, such that $T$ reflects a vector about the line $y = -x$.

What I did was take the vectors

\begin{bmatrix} 0 \\ 1 \end{bmatrix} and \begin{bmatrix} 1 \\ 0 \end{bmatrix}

and drew the corresponding images and performed the transformation and got the matrix \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}

However, the answer given is \begin{bmatrix} 0 & -1 \\ -1 & 0 \end{bmatrix}

This lead me to wonder, is the standard matrix of the transformation not unique, or is my attempt at solving this completely wrong? Clarification would be greatly appreciated.

Answer 1

This is the correct answer $$\begin{bmatrix} 0 & -1 \\ -1 & 0 \end{bmatrix}$$

The first column is the reflection of $(1,0)$ and the second column is the reflection of $(0,1)$.