Homework 17: Gram-Schmidt Orthogonalization, Determinants

Instructions. Problem 5 is optional.

1. Section 6.3, #10, 14, 37, 42(b), 43.
2. Section 2.1, #22, 28, 30.
3. Section 2.3, #8.
4. Section 5.1, #3, 4, 33. For all of these problems, you should not use determinants.
5. Let \(T:\bR^3\to \bR^3\) be the linear transformation given by reflecting across the plane \(x_1-2x_2+2x_3=0\). The goal of this problem is to find the standard matrix for \(T\).
   1. Find an orthogonal basis \(B=\{\bv_1,\bv_2,\bv_3\}\) for \(\bR^3\) such that \(\bv_1,\bv_2\) span the plane. There are several ways to do this. Here is one:
      • 1. Find a vector \(\bv_3\) which is orthogonal to all vectors on the plane. (This is covered in MATH 223. No calculations are required.)
      • 2. Find two vectors \(\bu_1,\bu_2\) which span the plane.
      • 3. Use the Gram-Schmidt process to replace \(\bu_1,\bu_2\) by two orthogonal vectors \(\bv_1,\bv_2\) which span the plane.
   2. Find \([T]_B\).
   3. Use the change of basis formula to find \([T]_{B'}\), where \(B'\) is the standard basis for \(\bR^3\).
      Answer: \( [T]_{B'}= \frac{1}{9} \begin{bmatrix} \pe 7 & \pe 4 & -4\\ \pe 4 & \pe 1 & \pe 8\\ -4 & \pe 8 & \pe 1 \end{bmatrix} \)
   4. Find the reflection of \(\vthree{\pe 1}{\pe 1}{-1}\) across the plane.