3.5 Orthonormal Basis

Definition 3.8 An orthonormal basis (ONB) in an \(n\)-dimensional vector space \(V\) consists of basis vectors \(\{\mathbf{b}_1, ..., \mathbf{b}_n\}\) that satisfy: \[ \langle \mathbf{b}_i, \mathbf{b}_j \rangle = 0 \text{ for } i \neq j, \quad \langle \mathbf{b}_i, \mathbf{b}_i \rangle = 1. \]

If only the first condition holds, the basis is orthogonal. The Gram-Schmidt process constructs an orthonormal basis from any set of linearly independent vectors.

Example 3.17 In \(\mathbb{R}^2\), \[ \mathbf{b}_1 = \frac{1}{\sqrt{2}}\begin{bmatrix}1 \\ 1\end{bmatrix}, \quad \mathbf{b}_2 = \frac{1}{\sqrt{2}}\begin{bmatrix}1 \\ -1\end{bmatrix} \] form an ONB because \(\mathbf{b}_1^T\mathbf{b}_2 = 0\) and \(\|\mathbf{b}_1\| = \|\mathbf{b}_2\| = 1\).

---

Exercises

Exercise 3.34

Show that \(\left\{\begin{bmatrix}1\\0\end{bmatrix}, \begin{bmatrix}0\\1\end{bmatrix} \right\}\) is an orthonormal basis.

Solution

Exercise 3.35

Show that \(\left\{\frac{1}{\sqrt{2}}\begin{bmatrix}1\\1\end{bmatrix}, \frac{1}{\sqrt{2}}\begin{bmatrix}1\\-1\end{bmatrix}\right\}\) is an orthonormal basis.

Solution

Exercise 3.36 Suppose I have 2 orthonormal basis vectors set up as columns of a matrix \(Q\). What is \(Q^TQ\)? What does this result imply about \(Q^T\)?

Solution

Exercise 3.37

Solution

Exercise 3.38

Solution