Can someone explain this linear transformation? I'm having a tough time understanding how this linear transformation works...

 A: In the left figure, the points $\begin{bmatrix} 0 \\ 2\end{bmatrix}$ and $\begin{bmatrix} 1 \\ 0\end{bmatrix}$ are pictured (using coordinates $\begin{bmatrix} x \\ y\end{bmatrix}$). Multiplying $\begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix}\begin{bmatrix} 0 \\ 2\end{bmatrix} = \begin{bmatrix} 2 \\ 0\end{bmatrix}$, and multiplying $\begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix}\begin{bmatrix} 1 \\ 0\end{bmatrix} = \begin{bmatrix} 0 \\ 1\end{bmatrix}$, so these are the two points pictured in the right figure.
The linear transformation being pictured is $T\left(\begin{bmatrix} x \\ y \end{bmatrix}\right) = \begin{bmatrix} y \\ x \end{bmatrix}$, which corresponds to left-multiplication by the given matrix, i.e.
\begin{equation}T\left(\begin{bmatrix} x \\ y \end{bmatrix}\right) = \begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix}\begin{bmatrix} x \\ y\end{bmatrix}.\end{equation}
A: Well, geometrically, this linear transformation acts as follows: 
 
Therefore, the linear transformation is in fact a composition of a reflection in the horizontal axis and a rotation along the negative vertical axis. 
