Suppose $U$ and $W$ are subspaces of $\mathbf R^8$ such that $\dim U = 3$, $\dim W = 5$, and $U + W = \mathbf R^8$. Prove that $\mathbf R^8 = U \oplus W$

By 2.43

Thus $U \cap W = \{0\}$, combine this with $U+W = \mathbf R^8$ to get $U \oplus W = \mathbf R^8$.