FALL 2000 FINAL EXAM FOR SM261
1330 DECEMBER 11, 2000

SHOW ALL WORK

Notations

$M_{m\times n}$ is the vector space of all $m\times n$ matrices

$P_n$ is the vector space of all polynomials of degree $\leq n$.

1. Consider the linear system of equations
   $$\begin{array}{cc} &3x+5y-4z=7\\ &-3x-2y+4z=-1\\ &6x+y+-8z=-4 \end{array}$$
   a. Write the system in the matrix form $Av=b$.

   b. Put the augmented matrix in reduced echelon form.

   c. Solve the system.

   d. Is the solution set a subspace of $\mathbb{R}^3$? Explain.

   e. Is the solution set of the corresponding homogenous system $Av=0$ a subspace of $\mathbb{R}^3$ Explain.

2. Find the inverse of the matrix
   $$\left( \begin{array}{ccc} 1&a&b\\ 0&2&c\\ 0&0&3 \end{array}\right)$$

3. For each of the following elementary row operations on a $2\times 2$ matrix $A$, find a matrix $E$ (called an elementary matrix) such that multiplying $A$ on the left by $E$ has the same result as the elementary row operation.

   a. Multiply row 2 by 4.

   b. Switch row 1 and row 2.

   c. Add 3 times row 2 to row 1.

4. Complete the definitions. $V$ is a vector space.

   a. A subset $S$ of $V$ is a subspace of $V$ if ....

   b. A set of vectors $\{v_1,\dots,v_n\}$ in $V$ is linearly independent if ....

   c. A set of vectors $\{v_1,\dots,v_n\}$ in $V$ spans $V$ if ....

   d. A set of vectors $\{v_1,\dots,v_n\}$ in $V$ is a basis for $V$ if ....

5. Let $V={\mathbb{R}}^4$, $$S=\left\{ \left( \begin{array}{c} x\\ y\\ z\\ w \end{array}\right) \in V\vert x+w=y+z\right\}$$. Determine if $S$ is a subspace of $V$. Justify your answer.

6. Define the linear transformation $T:\mathbb{R}^3\to\mathbb{R}^4$ by
   $$T \left( \begin{array}{c} a\\ b\\ c \end{array}\right) = \left( \begin{array}{c} a+b\\ a+c\\ b+c\\ a+b+c \end{array}\right).$$
   Recall that the range of $T$ is defined to be $\{T(v)\vert v\in\mathbb{R}^3\}$. The range of $T$ is a subspace of $\mathbb{R}^4$. Find a basis $\cal B$ for the range of $T$.

7. a. Let
   $$S=\{ \left( \begin{array}{c} x\\ y\\ z\\ w \end{array}\right) \in\mathbb{R}^4\vert x-y+2z-3w=0 \ {\rm and\ } y-z+w=0 \}.$$
   $S$ is a subspace of $\mathbb{R}^4$. Find a basis for $S$.

8. Let $V=\{A\in M_{4\times 4}\vert A{\rm\ is\ upper\ triangular }\}$. $V$ is a vector space. Find the dimension of $V$.

9. True or false? If the statement is false, give a counterexample.

   a. If a set of three vectors is linearly dependent, then one of the three vectors is a multiple of one of the other two vectors.

   b. If $v$ and $w$ are eigenvectors for different eigenvalues, then $v$ and $w$ are linearly independent.

   c. If $v$ and $w$ are eigenvectors for the same eigenvalue, then $v$ and $w$ are linearly dependent.

   d. An $n\times n$ matrix is diagonalizable if it has at least $n$ eigenvectors.

   e. An $n\times n$ matrix is diagonalizable if it has $n$ distinct eigenvalues.

10. Let $V=P_2$. Let $S$ be the subspace of $V$ with basis ${\cal B}=\{1-t,2t+t^2\}$.

    a. Let $w=2+8t+5t^2$. Show $w\in S$.

    b. Find $[w]_{\cal B}$, the coordinate vector of $w$ relative to $\cal B$.

11. Let $T:P_2\to P_3$ be the linear transformation given by
    $$T(p(t))=(3+2t)p(t)$$
    Find the matrix of $T$ relative to the ordered bases $\{1+t,1+t^2,t+t^2\}$ for $P_2$ and $\{1,t,t^2,t^3\}$ for $P_3$.

12. Let
    $$A=\left( \begin{array}{cc} 7&4\\ -3&-1 \end{array}\right).$$
    a. Find the characteristic polynomial of $A$.

    b. Find the eigenvalues of $A$.

    c. Find a basis for each eigenspace of $A$.

    d. Diagonalize $A$ by finding an invertible matrix $P$ and a diagonal matrix $D$ such that $A=PDP^{-1}$.

    e. Find a basis $\cal B$ for $\mathbb{R}^2$ relative to which the matrix of the linear transformation $T(v)=Av$ is a diagonal matrix. Find the matrix of $T$ relative to this basis $\cal B$.

13. Let $$A=\left( \begin{array}{cc} .8&.4\\ -.9&.8\end{array}\right)$$.

    a. Find a $2\times 2$ matrix $C$ which is similar to $A$ and such that the linear transformation $T(v)=Cv$ is a rotation of $\mathbb{R}^2$.

    b. Find the angle of rotation which is defined by $T$.

    c. Give an example of a $2\times 2$ matrix which has no real eigenvalues and which is not similar to a rotation.