Is every direct sum of subpaces orthogonal?

Question

Let $V$ be a finite dimensional vector space and let $V_1$ be a subspace of $V$ . Suppose that there exists a subspace $V_2$ of $V$ such that $V$ is the direct sum of $V_1$ and $V_2$: $$V= V_1 \oplus V_2.$$

Can we find an inner product on $V$ such that $V_1$ and $V_2$ are orthogonal ?

Answer 1

Assuming that they are real vector spaces we can. Pick bases $\{u_1,u_2,\dots,u_k\}$ for $V_1$ and $\{u_{k+1},u_{k+2},\dots,u_{n}\}$ for $V_2$. Their union is a basis for $V$ and the inner product $$ \left\langle \sum_1^n a_i u_i, \sum_1^n b_i u_i \right\rangle = \sum_1^n a_ib_i $$ makes them orthogonal.