Answer
W is a vector subspace of $R^3$.
Work Step by Step
Let $W=\left\{\left(x_{1}, x_{2}, 0\right) : x_{1} \text { and } x_{2} \text { are real numbers }\right\}$ and $u=(x_{1}, x_{2}, 0), v=(y_{1}, y_{2}, 0)\in W$, $c\in R$. First, one can see that $W$ contains the zero vector, and
$$u+v=(x_{1}, x_{2}, 0)+(y_{1}, y_{2}, 0)=(x_{1}+y_{1}, x_{2}+y_{2}, 0)\in W,$$
$$cu=c(x_{1}, x_{2}, 0)=(cx_{1}, cx_{2}, 0)\in W.$$
Hence, W is a vector subspace of $R^3$.
