I am trying to find a basis for the set of all $n \times n$ matrices with trace $0$. I know that part of that basis will be matrices with $1$ in only one entry and $0$ for all others for entries outside the diagonal, as they are not relevant.
I don't understand though how to generalize for the entries on the diagonal. Maybe just one matrix with $1$ in the $(1, 1)$ position and a $-1$ in all other $n – 1$ positions?
Best Answer
The matrix unit $E_{ij}$ is the matrix with $1$ in the $(i, j)$-entry and $0$ everywhere else. A basis for your space consists is $$ \{ E_{ij} \; \mid \; i \ne j \} \cup \{ E_{ii} - E_{i+1, i+1} \; \mid \; 1 \le i < n \}. $$
Notice that there are $n(n-1)$ of the off diagonal matrices and $n-1$ of the diagonal ones, for a total of $n^2 - 1$ matrices. This is the right size set since your space is the null space of the onto map $$ \operatorname{tr}: \Bbb{R}^{n^2} \to \Bbb{R}. $$