How exactly would I do this? I know for W to be a subspace of M22 it has to be a vector space under the operations of addition and scalar multiplcation, but how exactly would I show this? :S
if W was a subspace, then any scalar multiple of an element of W would have to be in W. in particular, -w has to be in W if w is. so if
$\displaystyle w=\begin{bmatrix}a&a+3\\0&b\end{bmatrix}$ we must have:
$\displaystyle \begin{bmatrix}-a&-a-3\\0&-b \end{bmatrix} \in W$. here, we have a problem, because -a-3 isn't 3 more than -a.
in fact, W fails all 3 tests for a subspace, it is not closed under matrix addition, or under scalar multiplication, nor does it contain the 0-matrix (choosing a= 0, or a=-3 doesn't work, we always get a non-zero entry).
closure tests are important, although checking to see that 0 is an element is the "easy one" (a vector space has to have a vector sum identity).