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

1.
Use Gauss-Jordan elimination to solve the following linear system.

x+2y-z = 4

2x+4y+3z = 5

-x-2y+6z = -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 P2 consisting of linear combinations of the three polynomials . First, show that 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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William W. Farr
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