**MA 2071 A '98 Final Exam Name:
.**

**Section Number: .**

Problem | 1 | 2 | 3 | 4 | 5 | 6 | Total |

Value | 10 | 20 | 10 | 20 | 20 | 20 | 100 |

Earned | |||||||

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

- 1.
- Use Gauss-Jordan elimination to solve the following linear
system.
*x*+2*y*-*z*= 42

*x*+4*y*+3*z*= 5-

*x*-2*y*+6*z*= -7 - 2.
- Explain the following terms or statements.
- (a)
- The set of vectors is linearly independent.
- (b)
- The set of vectors is a basis for a vector space
*V*. - (c)
- Rank of an matrix
*A*. - (d)
*L*is a linear transformation from to .- (e)
- The matrix
*A*is invertible.

- 3.
- One of the following statements about an matrix is
not equivalent to the
others. Find the one that is not equivalent and explain why it isn't.
- (a)
*A*is nonsingular.- (b)
- only has the trivial solution.
- (c)
- A has rank
*n*. - (d)
- Zero is an eigenvalue of
*A*.

- 4.
- Suppose that
*L*is the function from into defined by*L*(*x*,*y*,*z*) = (*x*+*y*,*x*-*y*,0). First, show that*L*is a linear transformation. Then compute the standard matrix representation of*L*. - 5.
- Let
*W*be the subset of*P*consisting of linear combinations of the three polynomials . First, show that_{2}*W*is a subspace, and then determine a basis for and the dimension of*W*. - 6.
- For the following matrix
*A*, find a matrix*P*, if possible, such that*P*^{-1}*A P*is a diagonal matrix. If it is not possible, explain why not.

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