Let $\displaystyle W_{1} $ and $\displaystyle W_{2}$ be subspaces of a vector space V. Let $\displaystyle W_{1}+W_{2} $ be the set of all vectors v in V such that $\displaystyle v=w_{1}+w_{2}$, where $\displaystyle w_{1} $is in $\displaystyle W_{1}$ and $\displaystyle w_{2}$ is in $\displaystyle W_{2}$. Show that $\displaystyle W_{1}+W_{2}$ is a subspace of V.