**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.

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

Then use your results to find a matrix such that is a diagonal matrix. - Use Gauss-Jordan elimination to solve the following linear
system.

- Suppose that is a linear transformation of
into . Given the following information,

compute , where

- Answer the following questions.
- Suppose that is a 3 by 5 matrix, and that the rank of is 2. Explain why this means that the dimension of the row space of is 2.
- Suppose that is the set of all ordered pairs which satisfy . Explain why cannot be a subspace of . Use the standard definitions for and .
- Suppose that is an matrix. Explain why the
following two statements are inconsistent.
- The rows of are linearly independent.
- Zero is an eigenvalue of A.

- The rows of are linearly independent.
- Explain why a set of four or more polynomials in must be linearly dependent.

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

2002-10-04