I have a problem that says:
Let $$V = \left\{ \begin{bmatrix} x\\ y\\ z\\ \end{bmatrix} \text{| $x +y+z = 0$} \right\} $$ $$W = \left\{ \begin{bmatrix} x\\ y\\ z\\ \end{bmatrix} \text{| $x +y = 0$} \right\} $$ Find a basis for $V$ and $W$, called $\alpha$ and $\beta$ respectively. Find $_\beta[T]_\alpha$, where $T : V \rightarrow W$ is defined by: $$T \left( \begin{bmatrix} x\\ y\\ z\\ \end{bmatrix} \right) = \begin{bmatrix} x - y\\ y + z\\ y - x\\ \end{bmatrix} $$
But is $T$ actually a transformation from $V$ to $W$? Because $(x-y) + (y + z) = x + z = -y$, which is not necessarily $0$. Am I just missing something here, or misunderstanding something?