Linear Algebra with Applications, 10th Edition

5.5 Orthonormal Sets

In ${ℝ}^2$, it is generally more convenient to use the standard basis $\left\{{\mathbf{e}}_1,{\mathbf{e}}_2\right\}$ than to use some other basis, such as $\left\{{\left(2,1\right)}^T,{\left(3,5\right)}^T\right\}$. For example, it would be easier to find the coordinates of ${\left(x_1,x_2\right)}^T$ with respect to the standard basis. The elements of the standard basis are orthogonal unit vectors. In working with an inner product space V, it is generally desirable to have a basis of mutually orthogonal unit vectors. Such a basis is convenient not only in finding coordinates of vectors, but also in solving least squares problems.

Definition

Let ${\mathbf{v}}_1,{\mathbf{v}}_2,\mathrm{\dots },{\mathbf{v}}_n$ be nonzero vectors in an inner product space V. If $\langle {\mathbf{v}}_i,{\mathbf{v}}_j\rangle =0$ whenever $i\ne j$, then $\left\{{\mathbf{v}}_1,{\mathbf{v}}_2,\mathrm{\dots },{\mathbf{v}}_n\right\}$ is said to be an orthogonal set of vectors.

Example 1

The set $\{{\left(1,1,1\right)}^{}$