Answer
$W$ is a subspace of $R^3$.
Work Step by Step
Let $W$ be a subset of $V$ such that
$$
W=\{(x, 2 x, 3 x) : x \text { is a real number }\}, \quad V=R^{3}.$$
Assume that $u=(a,2a,3a), v=(b,2b,3b)\in W$ and $c\in R$. Now, we have
(a) $W$ contains the zero vector $(0,0,0)$.
(b) \begin{align*}
u+v&=(a,2a,3a)+(b,2b,3b)\\
&=(a+b,2a+2b,3a+3b)\\
&=(a+b,2(a+b),3(a+b))\in W.
\end{align*}
(c) $cu=c(a,2a,3a)=(ca,2ca,3ca)\in W$.
Hence, $W$ is a subspace of $R^3$.
