Answer
As there are 2 pivot columns the dimension is 2.
Work Step by Step
Given Subspace:
$\begin{bmatrix}1\\0\\2\end{bmatrix},\begin{bmatrix}3\\1\\1\end{bmatrix},\begin{bmatrix}9\\4\\-2\end{bmatrix},\begin{bmatrix}-7\\-3\\1\end{bmatrix}$
Let's write the given vectors in augmented matrix $\mathbf{M}$ form and row reduce to echelon form;
$\mathbf{M}=\begin{bmatrix}1&3&9&-7\\0&1&4&-3\\2&1&-2&1\end{bmatrix}\sim\begin{bmatrix}1&3&9&-7\\0&1&4&-3\\0&0&0&0\end{bmatrix}$
As there are 2 pivot columns the dimension is 2.
