Fall 2001 Final exam for SM 261

December 10, 2001

Show all work. (Do row reductions in problem 1 by hand. All other row reductions can be done on the calculator.)

1. Consider the linear system of equations
   

   $$3 x + 5 y - 4 z = 7$$

   $$- 3 x - 2 y + 4 z = - 1$$

   $$6 x + y - 8 z = - 4$$

   
   

   a)

   If we write the system in the form $Av=b$, what are $A$, $b$?

   b)

   Put the augmented matrix in reduced row echelon form. Do by hand and show all steps.

   c)

   Solve the system.

   d)

   Is the solution set a subspace of $\mathbb{R}^3$? Explain. If so, what is its dimension?

   e)

   Is the solution set of the corresponding homogeneous syetm $Av=0$ a subspace of $\mathbb{R}^3$? Explain. If so, what is its dimension?

2. $T$ is a linear transformation from $\mathbb{R}^3$ to $\mathbb{R}^2$ such that $T\left( \begin{array}{c} 1 \\ 0 \\ 0 \end{array}\right) =\left( \begin{array}{c} 2 \\ 3 \end{array}\right)$, $T \left( \begin{array}{c} 0 \\ 1 \\ 0 \end{array}\right) = \left( \begin{array}{c} 1 \\ 1 \end{array}\right)$ , $T\left( \begin{array}{c} 1 \\ 1 \\ 1 \end{array}\right) = \left( \begin{array}{c} 4 \\ 5 \end{array}\right)$. Find the matrix of $T$.

3. Let $V$ be the subspace of $\mathbb{R}^4$ consisting of all solutions to the equation $x-2y+3z-w=0$. Find a basis for $V$.

4. Let $v_1=(1,2,0,1)$, $v_2=(-1,0,3,0)$, $v_3=(4,1,2,1)$. Let $V=span\{v_1,v_2,v_3\}$.

   a)

   Is $(9,7,-4,4)$ in $V$?

   b)

   Find dim($V$). Give your reasoning.

5. Let
   $$A=\left( \begin{array}{cccccc} 1 & 1 & 4 & -3 & -2 & 0\\ 2 & 3 ... ...3 & 1\\ -1 & 3 & -2 & 5 & 0 & 4\\ 3 & 5 & 5 & 0 & 1 & 2 \end{array}\right).$$

   a)

   Find a basis for $ker(A)$.

   b)

   Find a basis for $im(A)$.

6. Consider the plane $2x-3y+4z=0$ with basis ${\cal B}=\{(8,4,-1),(5,2,-1)\}$.

   a)

   Let $v=(1,-2,-2)$. Find $[v]_{\cal B}$, the ${\cal B}$-coordinate vector of $v$.

   a)

   If $[w]_{\cal B}=(1,-2)$, find $w$.

7. Find an orthonormal basis of $\mathbb{R}^3$ which contains the vector $(1/\sqrt{2},0,1/\sqrt{2})$.

8. Let $V$ be a subspace of $\mathbb{R}^4$ with basis $v_1=(1/\sqrt{2},0,0,1/\sqrt{2})$, $v_2=(0,1/\sqrt{2},1/\sqrt{2},0)$.

   a)

   Find the matrix of an orthogonal projection of $\mathbb{R}^4$ onto $V$.

   a)

   Find the orthogonal projection of $(1,3,5,2)$ onto $V$.

9. True or false. If false, give a counterexample.

   a)

   If $T:\mathbb{R}^n\rightarrow \mathbb{R}^n$ is the orthogonal projection onto a subspace $V$ of $\mathbb{R}^n$ then $T$ is an orthogonal transformation.

   a)

   The set of all eigenvectors of a $3\times 3$ matrix is a subspace of $\mathbb{R}^3$.

   a)

   Every set of $3$ vectors in $\mathbb{R}^4$ spans a subspace of $\mathbb{R}^4$.

10. Let
    $$A=\left( \begin{array}{cccccc} 1 & 0 & 0 & 5\\ 0 & 2 & 0 & 0 \\ 0 & 0 & 3 & 0\\ 6 & 0 & 0 & 4 \end{array}\right).$$

    a)

    Find the characteristic polynomial $f_A(\lambda)$ of $A$.

    a)

    Show that the eigenvalues of $A$ are $2$, $3$. Find the multiplicity of each eigenvalue.

    a)

    For each eigenvalue $\lambda$, find a basis for the eigenspace $E_\lambda$.

    a)

    Find the geometric multiplicity of each eigenvalue.

    a)

    Is $A$ diagonalizable using real matrices? Is $A$ diagonalizable using complex matrices? Give your reasoning.

11. Suppose that $v$ is an eigenvector for $A$ with eigenvalue $3$. Show that $v$ is also an eigenvector for the matrix $B=4A+5I$. What is the eigenvalue associated to $v$ with $v$ is considered as an eigenvector of $B$?