294.final.03

# 294.final.03 - Math 294 Final exam. Fall 2003. Do All 6...

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Math 294 Final exam. Fall 2003. Do All 6 problems. Your must justify all your answers except for problem 6 which consists of a number of true or false questions. Points will be deducted for illegible writing. No calculators. 1a Find an orthornormal eigenbasis for the linear transformation 101 010 A = . (20pts) 1b Find m A x G , where 1 0 0 x ⎛⎞ ⎜⎟ = ⎝⎠ G and m is a positive integer. Express your answer in m . (10pts) 1c Solve dx Ax dt = G G , with initial condition 1 (0 ) 0 0 xt == G . (10pts) 1d For a n by n matrix A , A e is defined as 2 lim ..... 1! 2! ! k A n k AA A eI k →∞ ≡+ + + + . Evaluate A e for A in (1a). (10pts) 2. Let ( ( )) (3 1) Tfx f x =− be a linear transformation from 2 P to 2 P . 2a. Find the matrix A of T with respect to the basis { } 2 1, , x x B= . (10pt) 2b. Find all the eigenvalues and eigenvectors of T . Is T diagonalizable ? (20pts) 3 Find a matrix B representating V proj x G , where x G is a vector in 3 R and V is the subspace of 3 R defined by 12 3 20 xx x −+ = . (21pts) 4a. Let N P

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## This note was uploaded on 06/10/2008 for the course MATH 2940 taught by Professor Hui during the Fall '05 term at Cornell.

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294.final.03 - Math 294 Final exam. Fall 2003. Do All 6...

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