Is the following a valid proof that $W$ is not a subspace of $V$? Advice on structuring the proof is welcome aswell.

$V = M_{m,n}$ is a vector space.

$W = \{x \in M_{m,n} : a_{ij} \ge 0\}$

In order for $W$ to be a subspace of $V$, it must be closed under addtion and scalar multiplication.

$W$ is closed under addition as:

$x, y \in W \Rightarrow x + y \in W$

$W$ fails to be closed under scalar multiplication as:

$(-1)x \notin W$

Thus $W$ is not a subspace of $V$.

## Best Answer

Two suggestions:

(1) I would say there is no need to state that $W$ is closed under addition, as that does nothing to help show that $W$ is not a subspace of $V$.

(2) (More important): I would give a specific example of a matrix $x\in W$ for which $(-1)x \notin W$. Otherwise one might note that the zero matrix, $z$, (all zeros) is in $W$ and so is $(-1)z$.