Linear Algebra (Math 220)
Assignment due Tuesday, March 11

1.  Preparation

Expect a quiz.

Suggested Reading:

• Lay § 4.7

• Hefferon §§ 3.IV – 3.V

2.  Exercises

1. Let $g$ be the linear map from ${\mathbf{R}}^4$ to ${\mathbf{R}}^4$ that is defined by $g\left(x\right)=Bx$ where $B$ is the matrix $$\left(\begin{array}{cccc}\hfill 1& \hfill 2& \hfill -4& \hfill 3\\ \hfill -2& \hfill -1& \hfill -1& \hfill 5\\ \hfill 1& \hfill 3& \hfill 2& \hfill -1\\ \hfill 1& \hfill 1& \hfill -1& \hfill -1\end{array}\right)\text{.}$$ Find a $4\times 4$ matrix $C$ for which the linear map $h$ given by multiplication by $C$ has the property that both $h\left(g\left(x\right)\right)=x$ and $g\left(h\left(y\right)\right)=y$ for all $x$ and all $y$ in ${\mathbf{R}}^4$.

2. Let $f$ be a linear map from ${\mathbf{R}}^3$ to ${\mathbf{R}}^3$ for which

   1. $f\left(1,0,0\right)=\left(1,2,3\right)$.

   2. $f\left(0,1⁄2,0\right)=\left(3,2,1\right)$.

   3. $f\left(-1,0,2\right)=\left(4,-6,2\right)$.

   Find all possible $3\times 3$ matrices $A$ for which the formula $f\left(x\right)=Ax$ is valid for all $x$ in ${\mathbf{R}}^3$.

   Hint: Use the rules for abstract linearity to work out what happens under $f$ to $\left(0,1,0\right)$ and $\left(0,0,1\right)$.

3. For a given real number $\theta$ find a $2\times 2$ matrix $R_{\theta }$ for which the linear function $\rho$ defined by $\rho \left(x\right)=R_{\theta }x$ is the counterclockwise rotation of the plane through the angle of (radian) measure $\theta$.

   Hint: First work out the four special cases where $\theta$ takes the values $0$, $\pi ⁄2$, $\pi$, and $3\pi ⁄2$.

4. Find a $3\times 3$ matrix $S$ for which the linear function $\sigma$ given by $\sigma \left(x\right)=Sx$ is the reflection of ${\mathbf{R}}^3$ in the $xz$ plane (where the $2^{\mathrm{nd}}$ coordinate $y=0$).