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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

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

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

x+2y-z = 4

2x+4y+3z = 5

-x-2y+6z = -7

Explain the following terms or statements.
The set of vectors $\{\mathbf{v}_1,\mathbf{v}_2, \ldots ,
\mathbf{v}_k\}$ is linearly independent.
The set of vectors $\{\mathbf{v}_1,\mathbf{v}_2, \ldots ,
\mathbf{v}_n\}$ is a basis for a vector space V.
Rank of an $n \times n$ matrix A.
L is a linear transformation from $\mathbf{R}^n$ to $\mathbf{R}^m$.
The $n \times n$ matrix A is invertible.

One of the following statements about an $n \times n$ matrix is not equivalent to the others. Find the one that is not equivalent and explain why it isn't.

A is nonsingular.
$A\mathbf{x} = \mathbf{0}$ only has the trivial solution.
A has rank n.
Zero is an eigenvalue of A.

Suppose that L is the function from $\mathbf{R}^3$ into $\mathbf{R}^3$ 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.

Let W be the subset of P2 consisting of linear combinations of the three polynomials $\{x+1, x^2, 2x+2\}$. First, show that W is a subspace, and then determine a basis for and the dimension of W.

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.

A = \left[ \begin{array}
 -8 & 18 \\  -3 & 7 
 \end{array} \right]\end{displaymath}

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William W. Farr