linear algebra - Does the set of matrix commutators form a subspace?

The following is an interesting problem from Linear Algebra 2nd Ed - Hoffman & Kunze (3.5 Q17).

Let $W$ be the subspace spanned by the commutators of $M_{n\times n}\left(F\right)$:
$$C=\left[A, B\right] = AB-BA$$
Prove that $W$ is exactly the subspace of matrices with zero trace.

Assuming this is true, one can construct $n^2 - 1$ linearly independent matrices, in particular

$$[e_{i,n}, e_{n,i}]\ \text{for $1\le i\le n-1$}$$
$$[e_{i,n}, e_{j,n}]\ \text{for $i\neq j$}$$
where $e_{i,j}$ are the standard basis with $0$ entry everywhere except row $i$ column $j$ which span the space of traceless matrices.

However, I have trouble showing (or rather, believing, since this fact seems to be given) that the set of commutators form a subspace. In particular, I am having difficulty showing that the set is closed under addition. Can anyone shed some light?

Answer

The set of matrix commutators is in fact a subspace, as every commutator has trace zero (fairly easy to prove) and every matrix with trace zero is a commutator (stated here but I know of no elementary proof), and the set of traceless matrices is clearly closed under linear combinations.

However, the problem is talking about the subspace spanned by the set of matrix commutators, which means the set of linear combinations of matrix commutators. This is by definition a subspace. This is probably because the proof that every matrix with trace zero is a commutator is difficult (although I'm not sure that this is the case).

Hope that clears things up a bit. If not, just ask!