Assignment #1

Announcement: One of the other faculty members will substitute on Monday, April 7, since I'll be out of town. For the same reason, I will not have office hours on Monday or Tuesday. Instead, I'll have an extra office hour Thursday, April 10, 2:30-3:30 (changed from a previous version).

The homework was supposed to be due on Wednesday, April 9, but this time it's OK to hand it in Friday, April 11 instead. Do look over the problems before the discussion section on Tuesday so that you can ask questions!

Reading: Chapters 1, 3. Look through Chapter 2 lightly; we may come back to this later.

To do but not hand in:

p. 11, Ex. 2;

C-1, C-2, C-4 [meaning from the Problems part of Handout C]

D-1 (below--do try this!)

To hand in:

p. 11, Ex. 5;

C-3;

D-2, D-3, D-4, D-5.

Problem D-1. Does the vector $(1,1,1)$ make a $45 ^\circ$ angle with the $z$-axis?

Problem D-2. (a) Let $M = \left[\begin{array}{rr}1&0\ 0&-1 \end{array}\right]$, a reflection in the $x$-axis. Show that $M R_{\theta} M = R_{-\theta}$, both by a calculation and by describing what the effect is if you do the left side to an object such as a piece of paper. (For clarity, imagine using paper with printing on it.)

(b) Show that $R_{-\theta} M = M R_{\theta}$. (Use (a), or calculate directly.)

(c) Show that the matrix of a reflection in R$^2$ whose mirror line makes an angle $\theta$ with the $x$-axis is $R _ {2 \theta} M$. (Start from the ``three-step method.'')

Problem D-3. There are only two $2 \times 2$ rotation matrices that have real eigenvalues. (a) Which ones? (b) Why are there no others? (Give some explanation.)

Problem D-4. Express a rotation of $45 ^\circ$ counterclockwise in $\mathbf{E}^ 2$ about the center $(1,1)$. Your answer should involve only one constant matrix and (if needed) one constant vector, with explicit entries. (Start by rewriting $(1,1)$ as a column vector. An explicit entry can be an expression, such as ${\frac 1 2}+ {\frac 1 2}\sqrt 3$. Leave in mathematical form rather than decimal approximations.)

Problem D-5. For the affine map $T: \mathbf{E}^ n \rightarrow \mathbf{E}^ n$ given by $T($p$) = A$p$+$   v, with $A$ nonsingular, show that the inverse map $T ^ {-1}$ is also affine and say explicitly what its linear and translational parts are in terms of $A$ and v.