Chapter 4 - Vector Spaces - 4.3 Subspaces of Vector Spaces - 4.3 Exercises - Page 167: 16

$W$ is not a vector subspace of $M_{3,1}$.

Let $u=\left[\begin{array}{lll}{2} & {0} & {\sqrt{2}}\end{array}\right]^{T}\in W$ and $c=2\in R$. Now, $$cu=2\left[\begin{array}{lll}{2} & {0} & {\sqrt{2}}\end{array}\right]^{T}=\left[\begin{array}{lll}{2} & {0} & {2\sqrt{2}}\end{array}\right]^{T}$$ which is not an element of $W$. Hence, $W$ is not a vector subspace of $M_{3,1}$.