Constructing an orthonormal basis for a subspace of $\mathcal M_{2\times2}$

Question

Let $S$ be a subspace of $\mathcal M_{2\times2}$ such that

$$S=\left \{\begin{bmatrix}a & b\\c & d\end{bmatrix}:a+b+c=0\right \}$$

Construct an orthonormal basis for S relatively to the inner product defined by $\langle A,B\rangle=\sum_{i,j=1}^2a_{ij}b_{ij}$.

I honestly don't really know where to begin, if I had simple vectors I'd just use the Gram-Schmidt process but working in a matrix subspace really confuses me...

Answer 1

It is easy to see that $S$ is a proper subspace of $\mathcal M_{2\times2}$ and that

$\begin{bmatrix}1 & -1\\0 & 0\end{bmatrix}, \begin{bmatrix}1 & 0\\-1 & 0\end{bmatrix}, \begin{bmatrix}0 & 0\\0 & 1\end{bmatrix} \in S.$

These three matrices are also linearly independent, hence

$\left \{\begin{bmatrix}1 & -1\\0 & 0\end{bmatrix}, \begin{bmatrix}1 & 0\\-1 & 0\end{bmatrix}, \begin{bmatrix}0 & 0\\0 & 1\end{bmatrix} \right \}$ is a basis of $S$.

Now proceed with Gram-Schmidt