## Orthogonal Nonzero Vectors Are Linearly Independent

## Problem 591

Let $S=\{\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_k\}$ be a set of nonzero vectors in $\R^n$.

Suppose that $S$ is an orthogonal set.

**(a)** Show that $S$ is linearly independent.

**(b)** If $k=n$, then prove that $S$ is a basis for $\R^n$.