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

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.

• The transformation depends on a choice of basis. – Karl Mar 19 '18 at 5:58
• It is unique, and the answer given is correct. By graphing this transformation, it should be clear that $(1, 0) \mapsto (0, -1)$ and $(0, 1) \mapsto (-1, 0)$. What you have produced is a rotation, counter-clockwise, by $\pi/2$ radians. – Theo Bendit Mar 19 '18 at 5:58
• A reflection should have a determinant of -1. So, the first answer can't possibly be correct, in any basis. – Joppy Mar 19 '18 at 10:07

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 $(0,1)$.