Homework 13 | Nicholas G. Vlamis

Homework 12
MATH 231
Complete before the final exam.

Instructions. We wiill have a quiz in class on the due date based on the content from the assignment. See the back of the textbook for solutions and hints for odd-numbered problems. Click here for a pdf version of the homework.

Exercise 1.

Complete the following exercises from Section 6.2 in the course textbook:

# 35, 37, 38

Exercise 2.

Complete the following exercises from Section 6.4 in the course textbook:

# 1, 7, 13, 24, 26

Exercise 3.

Complete the following exercises from Section 6.5 in the course textbook:

# 9, 15, 32

Exercise 4.

Let $A$ be an $n\times n$ orthogonal matrix. Show that if $\mathbf{u},\mathbf{v}\in\mathbb{R}^{n}$ , then

(A\mathbf{u})\cdot(A\mathbf{v})=\mathbf{u}\cdot\mathbf{v}.

(This says that orthogonal matrices, as linear transformations, preserve distance. As they also fix the origin, they must fix the sphere, which is made up of unit vectors. So, it induces an isometry of the $(n-1)$ -sphere.)

Exercise 5.

Let $R_{\theta}=\begin{bmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{bmatrix}$ . Show that $R_{\theta}$ is an orthogonal matrix.