Homework 16: Inner Product Spaces

1. Section 6.1, #36, 44.
2. Section 6.2, #37, 39, 40, 41, 42.
3. Let \(T:\bR^3\to \bR^3\) be rotation by \(\theta\) about the line through \(\vthree{0}{1}{1}\) (counterclockwise, as viewed from the tip of \(\vthree{0}{1}{1}\)). Let \(\cB\) be the the following basis of \(\bR^3\): $$ \cB= \left\{ \vthree{1}{0}{0}, \vthree{0}{\frac{1}{\sqrt{2}}}{\frac{-1}{\sqrt{2}}}, \vthree{0}{\frac{1}{\sqrt{2}}}{\frac{1}{\sqrt{2}}} \right\}. $$ Let \(\cB'=\{\be_1,\be_2,\be_3\}\), the standard basis for \(\bR^3\).
1. Verify that \(\cB\) is an orthonormal basis for \(\bR^3\).
2. Find \([T]_{\cB}\).
3. Use the change of basis formula to find \([T]_{\cB'}\), the standard matrix for \(T\).
4. Using (c), find the rotation of \(\vthree{3}{4}{5}\) by \(\pi/3\) about the line through \(\vthree{0}{1}{1}\).
Note. For this problem, you should use the Euclidean inner product (i.e., the dot product) on \(\bR^3\).