Chapter 5.6, Problem 15E

(Dimension Theorem) Let U and V be subspaces of ${ℝ}^n.$ In the case that $U\cap V=\left\{0\right\},$ we have the following dimension relation $\dim \left(U+V\right)=\dim U+\dim V$ (See Exercise 18 in Section 3.4 of Chapter 3.) Make use of the result from Exercise 14 to prove the more general theorem
$\dim \left(U+V\right)=\dim U+\dim V-\dim \left(U\cap V\right)$