Answer
$W$ is a vector subspace of $R^3$.
Work Step by Step
Let $u=(x_1,x_2,2x_1-3x_2),v=(y_1,y_2,2y_1-3y_2),\in W$ and $c\in R$ where
$$W=\{(x, y, 2 x-3 y) : x \text { and } y \text { are real numbers }\}$$
Now,
\begin{align*}
u+v&=(x_1,x_2,2x_1-3x_2)+(y_1,y_2,2y_1-3y_2)\\
&=(x_1+y_1,x_2+y_2,2(x_1+y_1)-3(x_2+y_2))
\end{align*}
which means that $u+v\in W$ and also $cu=(cx_1,cx_2,c(2x_1-3x_2))\in W$. Hence, $W$ is a vector subspace of $R^3$.
