Exercises

1.
The vectors u = $\left[\vphantom{
\begin{array}{ccc}
1 & 1 & 0
\end{array}
}\right.$$\begin{array}{ccc}
1 & 1 & 0
\end{array}$$\left.\vphantom{
\begin{array}{ccc}
1 & 1 & 0
\end{array}
}\right]$ and v = $\left[\vphantom{
\begin{array}{ccc}
1 & 1 & 1
\end{array}
}\right.$$\begin{array}{ccc}
1 & 1 & 1
\end{array}$$\left.\vphantom{
\begin{array}{ccc}
1 & 1 & 1
\end{array}
}\right]$ span a plane in $\Bbb$R3. Find the projection matrix P onto the plane, and find a nonzero vector b that is projected to zero. BITMAPSETAnswer0.2214in0.205in0ina1

2.
For the following matrix, find the characteristic polynomial, minimum polynomial, eigenvalues, and eigenvectors. Discuss the relationships among these, and explain the multiplicity of the eigenvalue.


$\displaystyle \left[\vphantom{
\begin{array}{cccc}
2 & 0 & 0 & 0 \\
1 & 2 & 0 & 0 \\
0 & 0 & 2 & 0 \\
0 & 0 & -3 & 2
\end{array}
}\right.$$\displaystyle \begin{array}{cccc}
2 & 0 & 0 & 0 \\
1 & 2 & 0 & 0 \\
0 & 0 & 2 & 0 \\
0 & 0 & -3 & 2
\end{array}$$\displaystyle \left.\vphantom{
\begin{array}{cccc}
2 & 0 & 0 & 0 \\
1 & 2 & 0 & 0 \\
0 & 0 & 2 & 0 \\
0 & 0 & -3 & 2
\end{array}
}\right]$

BITMAPSETAnswer0.2214in0.205in0ina2

3.
Which of the following statements are correct for the matrix A = $\left[\vphantom{
\begin{array}{ccc}
1 & 1 & 1 \\
1 & 0 & 2
\end{array}
}\right.$$\begin{array}{ccc}
1 & 1 & 1 \\
1 & 0 & 2
\end{array}$$\left.\vphantom{
\begin{array}{ccc}
1 & 1 & 1 \\
1 & 0 & 2
\end{array}
}\right]$? The set of all solutions x = $\left[\vphantom{
\begin{array}{c}
x_{1} \\
x_{2} \\
x_{3}
\end{array}
}\right.$$\begin{array}{c}
x_{1} \\
x_{2} \\
x_{3}
\end{array}$$\left.\vphantom{
\begin{array}{c}
x_{1} \\
x_{2} \\
x_{3}
\end{array}
}\right]$ of the equation Ax = $\left[\vphantom{
\begin{array}{c}
0 \\
0
\end{array}
}\right.$$\begin{array}{c}
0 \\
0
\end{array}$$\left.\vphantom{
\begin{array}{c}
0 \\
0
\end{array}
}\right]$ is the column space of A; the row space of A; a nullspace of A; a plane; a line; a point.BITMAPSETAnswer0.2214in0.205in0ina3

4.
The matrices that rotate the xy-plane are A$\left(\vphantom{
\theta }\right.$θ$\left.\vphantom{
\theta }\right)$ = $\left[\vphantom{
\begin{array}{rr}
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta
\end{array}
}\right.$$\begin{array}{rr}
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta
\end{array}$$\left.\vphantom{
\begin{array}{rr}
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta
\end{array}
}\right]$. Verify that A$\left(\vphantom{
\theta }\right.$θ$\left.\vphantom{
\theta }\right)$A$\left(\vphantom{ \varphi }\right.$$\varphi$$\left.\vphantom{ \varphi }\right)$ = A$\left(\vphantom{ \theta +\varphi }\right.$θ + $\varphi$$\left.\vphantom{ \theta +\varphi }\right)$ and A$\left(\vphantom{ -\theta }\right.$ - θ$\left.\vphantom{ -\theta }\right)$ = A$\left(\vphantom{
\theta }\right.$θ$\left.\vphantom{
\theta }\right)^{{-1}}_{}$, using matrix products and trigonometric identities.


BITMAPSETAnswer0.2214in0.205in0ina4



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