Linear Algebra (Math 220)
Assignment due Thursday, April 24

1.  Preparation

Expect a quiz.

Relevant Reading:

Lay §§ 5.1 – 5.3

Hefferon §§ 5.I – 5.II

2.  Exercises

1. When $\mathbf{h}$ is the basis of the Cartesian plane with $h_1=\left(a,b\right)$ and $h_2=\left(c,d\right)$, what is the matrix of the rotation about the origin through the angle $\pi ⁄2$ relative to $\mathbf{h}$? (Assume that $ad-bc\ne 0$.)

2. Let $f$ be the linear function from ${\mathbf{R}}^3$ to ${\mathbf{R}}^3$ that has the matrix $$D=\left(\begin{array}{ccc}\hfill 2& \hfill 0& \hfill 0\\ \hfill 0& \hfill 1& \hfill 0\\ \hfill 0& \hfill 0& \hfill 3\end{array}\right)$$ relative to the basis of ${\mathbf{R}}^3$ given by the columns of the matrix $$\left(\begin{array}{ccc}\hfill 3& \hfill 6& \hfill 2\\ \hfill 2& \hfill -3& \hfill 6\\ \hfill 6& \hfill -2& \hfill -3\end{array}\right)\text{.}$$

   1. How many lines $L$ passing through the origin have the properly that $f$ carries each point of $L$ to a point of $L$ ?

   2. Find all points $x$ in ${\mathbf{R}}^3$ for which $f\left(x\right)=x$.

   3. For each of two different lines through the origin find a point on the line that is carried to another point on the same line.

3. Let $S$ be the $2\times 2$ matrix $$\left(\begin{array}{cc}\hfill 3⁄5& \hfill 4⁄5\\ \hfill 4⁄5& \hfill -3⁄5\end{array}\right)\text{.}$$

   1. Find a point $P$ in ${\mathbf{R}}^2$ at distance $1$ from the origin for which $SP=-P$.

   2. Find a line in ${\mathbf{R}}^2$ characterized by the property that the matrix $S$ represents the reflection in that line relative to the standard basis of ${\mathbf{R}}^2$.

   3. Find an orthogonal matrix $U$ for which $$U^{-1}SU$$ is a diagonal matrix.