Chapter 3.3, Problem 69E

Consider two subspaces V and W of ${ℝ}^n$ . Show that $\text{dim}\left(V\right)+\text{dim}\left(W\right)=\dim \left(V\cap W\right)+\dim \left(V+W\right)$ .For the definition of $V+W$ , see Exercise 3.2.50.Hint: Pick a basis ${\stackrel{\to }{u}}_1,{\stackrel{\to }{u}}_2,\mathrm{...}{\stackrel{\to }{u}}_m$ of $V\cap W$ . UsingExercise 67, construct ${\stackrel{\to }{u}}_1,{\stackrel{\to }{u}}_2,\mathrm{...}{\stackrel{\to }{u}}_m,{\stackrel{\to }{v}}_1,{\stackrel{\to }{v}}_2,\mathrm{...}{\stackrel{\to }{v}}_p$ of V and ${\stackrel{\to }{u}}_1,{\stackrel{\to }{u}}_2,\mathrm{...}{\stackrel{\to }{u}}_m,{\stackrel{\to }{w}}_1,{\stackrel{\to }{w}}_2,\mathrm{...}{\stackrel{\to }{w}}_q$ of W.Show that ${\stackrel{\to }{u}}_1,{\stackrel{\to }{u}}_2,\mathrm{...}{\stackrel{\to }{u}}_m,{\stackrel{\to }{v}}_1,{\stackrel{\to }{v}}_2,\mathrm{...}{\stackrel{\to }{v}}_p,{\stackrel{\to }{w}}_1,{\stackrel{\to }{w}}_2,\mathrm{...}{\stackrel{\to }{w}}_q$ is a basis of $V+W$ . Demonstrating linear independence is somewhat challenging.