1502JTest3

# 1502JTest3 - ˜ ˜ 4(25 The matrix A = 1 1-4 4 2 2-1 1 ˆ...

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Name±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±± 15 April 2004 Teaching Assistant±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±± Hour Test 3 Math 1502J 2pm Andrew Instructions: 1. Closed book. 2. Show your work and explain your answers and reasoning. 3. Calculators may be used, but pay particular attention to instruction 2. To receive credit, you must show your work. Unexplained answers, and answers not supported by the work you show, will not receive credit. 4. Express your answers in simplified form. 1. (25) Use the Normal Equations method to find the least squares solution to Ax = b where A = 1 1 2 0 - 1 2 ˆ ˜ ˜ ˜ and b = - 6 4 11 ˆ ˜ ˜ ˜ . 2. (25) Find a basis for the Image and a basis for the Kernel of the matrix A = 1 3 1 4 2 0 - 4 2 1 1 - 1 0 1 1 - 1 0 ˆ ˜ ˜ ˜ ˜ . 3. (25) Use the cross product to find the area of the triangle with vertices at P = 1 1 4 ˆ ˜ ˜ ˜ , Q = 1 2 6 ˆ ˜ ˜ ˜ , and R = - 1 4 2 ˆ ˜

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Unformatted text preview: ˜ ˜ . 4. (25) The matrix A = 1 1-4 4 2 2-1 1 ˆ ˜ ˜ ˜ has characteristic polynomial 1 -t ( ) 2 -t ( ) 2 . a. Find all eigenvalues and eigenvectors of A . b. Either find an invertible matrix S and a diagonal matrix D with S-1 AS = D or explain why no such S and D exist. Name±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±± Page 2 of 2 Math 1502J 2pm Andrew Hour Test 3 Teaching Assistant±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±± 15 April 2004 Answers. 1. -1 3 ˆ ˜ 2. A basis for the kernel is 2-1 1 ˆ ˜ ˜ ˜ ˜ . A basis for the image consists of the pivot columns 1 2 1 1 ˆ ˜ ˜ ˜ ˜ , 3 1 1 ˆ ˜ ˜ ˜ ˜ , 4 2 ˆ ˜ ˜ ˜ ˜ 3. 21 4. a. l = 1 , -1-2 1 ˆ ˜ ˜ ˜ and = 2 , 1 2 ˆ ˜ ˜ ˜ , 1 2 ˆ ˜ ˜ ˜ b. S =-1 1 1-2 2 1 2 ˆ ˜ ˜ ˜ D = 1 2 2 ˆ ˜ ˜ ˜...
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## This note was uploaded on 04/07/2008 for the course MATH 1502 taught by Professor Mcclain during the Fall '07 term at Georgia Tech.

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1502JTest3 - ˜ ˜ 4(25 The matrix A = 1 1-4 4 2 2-1 1 ˆ...

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