Answer
$W$ is a vector subspace of $M_{2.2}$.
Work Step by Step
Let $u=\left[\begin{array}{ll}{0} & {a} \\ {b} & {0}\end{array}\right],v=\left[\begin{array}{ll}{0} & {c} \\ {d} & {0}\end{array}\right]\in W$ and $c'\in R$ where $W$ the set of all $2\times 2$ matrices of the form $\left[\begin{array}{ll}{0} & {a} \\ {b} & {0}\end{array}\right]$.
Now,
\begin{align*}
u+v&=\left[\begin{array}{ll}{0} & {a} \\ {b} & {0}\end{array}\right]+\left[\begin{array}{ll}{0} & {c} \\ {d} & {0}\end{array}\right]\\
&=\left[\begin{array}{ll}{0} & {a+c} \\ {b+d} & {0}\end{array}\right]
\end{align*}
which means that $u+v\in W$ and also $c'u=\left[\begin{array}{ll}{0} & {c'a} \\ {c'b} & {0}\end{array}\right]\in W$. Hence, $W$ is a vector subspace of $M_{2.2}$.
