## Elementary Linear Algebra 7th Edition

$W$ is a vector subspace of $R^4$.
Let $u=(x_1,x_2,x_3,0),v=(y_1,y_2,y_3,0),\in W$ and $c\in R$ where $$W=\left\{\left(x_{1}, x_{2}, x_{3}, 0\right) : x_{1}, x_{2}, \text { and } x_{3} \text { are real numbers }\right\}.$$ Now, \begin{align*} u+v&=(x_1,x_2,x_3,0)+(y_1,y_2,y_3,0)\\ &=(x_1+y_1,x_2+y_2,x_3+y_3,0) \end{align*} which means that $u+v\in W$ and also $cu=(cx_1,cx_2,cx_3,0)\in W$. Hence, $W$ is a vector subspace of $R^4$.