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Final-Fall-2005-06-Mouzeihem-and-Houri-and-Audi

# Final-Fall-2005-06-Mouzeihem-and-Houri-and-Audi - AMERICAN...

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AMERICAN UNIVERSITY OF BEIRUT Mathematics Department Math 218 - Final Exam Fall 2005-2006 Name: .................................. ID: ................................ Section 1 Section 2 Section 3 Mrs. Z. Mouzeihem Ms. M. Houri Ms. D. Audi Time: 120 min I- ( 35 points ) Let A = 1 2 1 1 3 7 2 2 2 4 2 2 (a) Find a basis for the row space of A. (b) Find a basis for the column space of A. 1

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(c) Find a basis for the null space of A . (d) Find a basis for the null space of A T . (e) If W is the column space of A deduce a basis for W. 2
II- ( 25 points ) Let T : P 2 P 2 be a linear transformation such that: T ( p ( x )) = p (2 x + 3) a) Find the matrix A = [ T ] B with respect to the standard basis B = { 1 , x, x 2 } . b) Find the eigenvalues for the matrix A = [ T ] B c) Write the formula for [ T ( p ( x ))] B without justification. 3

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d) Deduce the values of λ such that T ( p ( x )) = λp ( x ) has nonzero solutions. III- ( 20 points ) Let S = { u 1 , u 2 , u 3 } be a basis for R 3 , with u 1 = (2 , 2 , 0) , u 2 = (2 , 0 , 0) and u 3 = (0 , 0 , 3) a) Find an orthonormal basis S 1 = { q 1 , q 2 , q 3 } for R 3 .
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Final-Fall-2005-06-Mouzeihem-and-Houri-and-Audi - AMERICAN...

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