Dimensions of null spaces of a matrix

Let $M=\begin{bmatrix}1&-1&1\\1&-1&-1\end{bmatrix}$. If

$$V=\left\{(x,y,0)\in\mathbb R^3:M\begin{bmatrix}x\\y\\0\end{bmatrix}=\begin{bmatrix}0\\0\end{bmatrix}\right\}\text{ and }W=\left\{(x,y,z)\in\mathbb R^3:M\begin{bmatrix}x\\y\\z\end{bmatrix}=\begin{bmatrix}0\\0\end{bmatrix}\right\},$$

then