MA 2071 A '01 Final Exam Name: .

Section Number:.




Problem 1 2 3 4 5 Total
Value 20 30 10 20 20 100
Earned            
             

Show your work in the space provided. Unsupported answers may not receive full credit.

  1. Find the eigenvalues and eigenvectors of the following matrix. You must show your work.

    \begin{displaymath}A = \left[ \begin{array}{cc}
2 &2 \\
2 &2
\end{array} \right]
\end{displaymath}

    Then use your results to find a matrix $P$ such that $P^{-1}AP$ is a diagonal matrix.

  2. Use Gauss-Jordan elimination to solve the following linear system.

    \begin{eqnarray*}
x_1+2x_2+3x_3 +2 x_4 - 4 x_5 & = & 14 \\
-2x_1+2x_2+5x_3 +7 x_4 - 10 x_5 & = & 17 \\
3x_1-x_2+2x_3 - x_4 + 9 x_5 & = & 7 \\
\end{eqnarray*}



  3. Suppose that $L$ is a linear transformation of $\mathbf{R}^2$ into $\mathbf{R}^3$. Given the following information,

    \begin{displaymath}L(\mathbf{e}_1) = \left[ \begin{array}{c}
-1 \\
2 \\
5
...
...left[ \begin{array}{c}
12 \\
-3 \\
7
\end{array} \right]
\end{displaymath}

    compute $L(\mathbf{u})$, where

    \begin{displaymath}\mathbf{u} = \left[ \begin{array}{c}
1 \\
2
\end{array} \right]
\end{displaymath}

  4. Answer the following questions.
    1. Suppose that $A$ is a 3 by 5 matrix, and that the rank of $A$ is 2. Explain why this means that the dimension of the row space of $A$ is 2.
    2. Suppose that $W$ is the set of all ordered pairs $(x,y)$ which satisfy $y < 0$. Explain why $W$ cannot be a subspace of $\mathbf{R}^2$. Use the standard definitions for $\oplus$ and $\odot$.
    3. Suppose that $A$ is an $n \times n$ matrix. Explain why the following two statements are inconsistent.
      • The rows of $A$ are linearly independent.

      • Zero is an eigenvalue of A.
    4. Explain why a set of four or more polynomials in $P_2$ must be linearly dependent.

  5. Find a basis for the column space of the matrix below.

    \begin{displaymath}A = \left[ \begin{array}{ccc}
1 & 3 & -2 \\
-1 & 2 & -3 \\
-1 & 3 & -4
\end{array} \right] \end{displaymath}

William W. Farr
2002-10-04