Use the Gram-Schmidt algorithm to compute an orthonormal basis.
Definition6.3.1.
A set \(S\) of vectors is called an orthogonal set provided \(\vec u \cdot \vec v = 0\) for any pair of vectors \(\vec u\text{,}\)\(\vec v \in S\) with \(\vec u \ne \vec v\text{.}\)
Activity6.3.2.
If we rotate the standard basis of \(\IR^2\) by \(45^{\circ}\text{,}\) we produce the set \(B=\left\{\vec b_1, \vec b_2\right\}=\left\{ \left[\begin{array}{c} \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} \end{array}\right], \left[\begin{array}{c} -\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} \end{array}\right]\right\}\)
(a)
Determine the length of \(\vec b_1\) and \(\vec b_2\text{.}\)
(b)
Determine the dot product \(\vec b_1 \cdot \vec b_2\text{.}\)
(c)
Is \(B\) is an orthogonal set? Explain why or why not.
(d)
Is \(B\) is a basis of \(\IR^2\text{?}\) Explain why or why not.
Definition6.3.3.
A basis \(B\) of a vector space is called an orthogonal basis provided every pair of vectors in \(B\) is orthogonal. If, in addition, each vector in \(B\) is a unit vector, then \(B\) is called an orthonormal basis.
