Chapter 3.3, Problem 66E

Two subspaces V and W of ${ℝ}^n$ arc called complementsif any vector $\stackrel{\to }{x}$ in ${ℝ}^n$ can be expressed uniquely as $\stackrel{\to }{x}=\stackrel{\to }{v}+\stackrel{\to }{w}$ , where $\stackrel{\to }{v}$ is in V and $\stackrel{\to }{w}$ is in W. Show that Vand W are complements if (and only if) $V\cap W=\left\{\stackrel{\to }{0}\right\}$ and $\text{dim}\left(V\right)+\text{dim}\left(W\right)=n$ .