Subspaces in linear algebra - Matrix

Question

Let $S=$ {$a_{ij} \in M_{3}(\mathbb{R}):a_{11}+a_{12}+a_{13}=a_{21}+a_{22}+a_{23}=a_{31}+a_{32}+a_{33}$}

$S$ is a subspace of $M_{3}(\mathbb{R})$ and dim $S = 7$

I tought I could arrive somewhere with the determinand but that led me nowhere. Any other ideas?

Answer 1

Yes. Take $V=\mathbb R^9$. Then consider the matrix $$ M=\begin{pmatrix} 1 & 1 & 1 & -1 & -1 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1& 1&1&-1&-1&-1 \end{pmatrix} $$

Then $S$ is the kernel of this matrix ($Mx=0$ if and only if $x_1+x_2+x_3 = x_4+x_5+x_6 = x_7+x_8+x_9$).

As $M$ has rank $2$, corank $7$ the kernel has dimension $7$.