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King Fahd University of Petroleum & Minerals Department of Math & Stat Math-280, Term-141 Class Test No. 4 11-12-2014 Name: ___________________________ ID #: ____________________ (Show all your work) 1) Given the map L : P3 P3 defined by L p x p x p x a) Show that L is a linear transformation. b) Find the kernel of L . c) Find the Range of L . /10 2 1 1 2) Let A and S e1 , e2 , e3 and T e1 , e2 3 2 1 R 3 and R2 , respectively. be the natural bases for /14 3 2 a) Find the linear transformation L : R R whose representation with respect to S and T is A . b) Let 1 S 0 , 1 1 1 , 0 0 1 and 1 1 T , 3 2 1 Be ordered bases for R 3 and R2 , respectively. Determine the linear transformation L : R 3 R 2 , whose representation with respect to S and T is A . c) 1 Compute L 2 using L as determined in part (b). 3 6 4 3) Given the Matrix A 3 1 a) Find the characteristic polynomial of A . b) Find the eigenvalues and the associated eigenvectors. c) Compute A 5 A 6 I 2 d) Show that if 0 is an eigenvalue of AB then, 0 is also an eigenvalue of BA . /14 0 2 4) Given the Matrix A 2 0 2 2 2 2 0 /12 a) Find a Matrix X that can Diagonalize A . Also, find the Diagonal Matrix D and its relation with X and A . b) Show that if A is diagonalizable, then AT is also diagonalizable.