Linear Algebra, 5th Edition

6.2 The Gram-Schmidt Orthogonalization Process and Orthogonal Complements

In previous chapters, we have seen the special role of the standard ordered bases for ${\text{C}}^n$ and ${\text{R}}^n$. The special properties of these bases stem from the fact that the basis vectors form an orthonormal set. Just as bases are the building blocks of vector spaces, bases that are also orthonormal sets are the building blocks of inner product spaces. We now name such bases.

Definition.

Let V be an inner product space. A subset of V is an orthonormal basis for V if it is an ordered basis that is orthonormal.

Example 1

The standard ordered basis for ${\text{F}}^n$ is an orthonormal basis for ${\text{F}}^n$.

Example 2

The set

$$\{\left({\displaystyle \frac{1}{\sqrt{5}}},{\displaystyle \frac{2}{\sqrt{5}}}\right),({\displaystyle \frac{2}{\sqrt{5}}},{\displaystyle \frac{-1}{\sqrt{5}}}$$