Suppose UUU and WWW are subspaces of R8\mathbf R^8R8 such that dimU=3\dim U = 3dimU=3, dimW=5\dim W = 5dimW=5, and U+W=R8U + W = \mathbf R^8U+W=R8. Prove that R8=U⊕W\mathbf R^8 = U \oplus WR8=U⊕W
By 2.43
Thus U∩W={0}U \cap W = \{0\}U∩W={0}, combine this with U+W=R8U+W = \mathbf R^8U+W=R8 to get U⊕W=R8U \oplus W = \mathbf R^8U⊕W=R8.