Solution 2b 8

Suppose $U$ and $W$ are subspaces of $V$ such that $V = U \oplus W$ . Suppose also that $u_1,\dots,u_m$ is a basis of $U$ and $w_1,\dots,w_n$ is a basis of $W$ .
Prove that

u_1,\dots,u_m,w_1,\dots,w_n

Is a basis of $V$ .

It's clearly spanning since $U+W = V$ , so we must show it's independent. Suppose

a_1u_1 + \dots + a_mu_m + b_1w_1 + \dots + b_nw_n = 0

Then

a_1u_1 + \dots + a_mu_m = -(b_1w_1 + \dots + b_nw_n)

The left side is in $U$ and the right side is in $W$ , since $U \cap W = \{0\}$ (because it's a direct sum) both sides must be zero, and since the $u$ 's and $w$ 's are independent among themselves this implies $a_k = 0$ and $b_k = 0$ for all $k$ . This completes the proof of independence, thus they are a basis of $V$ .