Chapter 4 - Vector Spaces - 4.3 Subspaces of Vector Spaces - 4.3 Exercises - Page 168: 53

The set $W$ is not a subspace of $ R^{3}$.

Let $A$ be a fixed $2\times3$ matrix and $x, y\in W$, then $Ax=[\begin{array} \\1 \\2 \end{array}]$ and $Ay=[\begin{array} \\1 \\2 \end{array}]$ Thus we have that $A(x+y)=Ax+Ay=[\begin{array} \\2 \\4 \end{array}]\ne [\begin{array} \\1 \\2 \end{array}] $ Therefore $ x+y \notin W$ Therefore the set $W$ is not a subspace of $ R^{3}$.