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University of Bahrain College of Science Department of Mathematics Summer Semester 2008/2009 Final Examination MATHS 211 Date : 24th August 2009 Max Marks: 50 Time: 08:30-10:30 Name: I.D. No: Marking Scheme Question Max. Marks 1 10 2 12 3 20 4 8 Total 50 Marks Obtained Good Luck 1 Section: QUESTION 1 [ 4,6 marks] a) Find the eigenvalues and basis of eigenspaces of 3 1 A 4 1 Is A diagonalizable? 2 b) Let T : M 22 P2 be linear transformation defined as a b a 2b c a c x a d x 2 T c d Find (i) a basis of kernel T. (ii) a basis of Range T. 3 QUESTION 2 [ 12 marks] a) Show whether the set A, B, C is linearly dependent or linearly independent in M 22 , where 1 2 0 1 1 1 A ,B ,C 0 1 3 4 0 2 4 b) Find the matrix X if 2 1 1 2 1 0 3 4 X 4 0 3 1 c) If A is n n matrix and det A 0 show whether the rows of A forms a linearly dependent or linearly independent set. 5 QUESTION 3 [20 marks] a) If T : P2 P3 is a linear transformation such that T ( x 2 ) x 3 , T x 1 1, T x 1 x Find T ( x 2 4 x 2) 1 1 b) Let W X in M 22 : AX X where A 0 0 Find a basis of W. 6 c) For the system x1 2 x2 x3 b1 x1 3x2 x3 b2 2 x1 4 x2 2 x3 b3 Find for what relation between b1 , b2 and b3 the system has no solution. d) Show that C 1 D 1 1 C (C D) 1 D 7 e) If A is 2 2 matrix, the eigenvalues of A-1 are 3 and 2 , the basis of 4 1 eigenspaces for 3 is and for 2 is 1 1 2 Find the matrix A . 8 QUESTION 4 [8 marks] In each part one of the 4 statements is correct, circle the correct statement a) (i) If A is diagonalizable then A-1 must exist. (ii) If A is m n m n then the rows or columns of A must be linearly dependent. (iii) If A is 5 4 then the system AX= 0 always have infinite solutions. (iv) If a matrix A is a square then rank A = nullity A. b) If S is a basis of V and S has n vectors then (i) any set of less than n vectors in V is always linearly dependent (ii) Any set of n vectors is also a basis of V. (iii) A vector u s is a vector in R n (iv) A set v1 , v2 , u does not span V for any vector u in V. c) If A is n n matrix then (i) A diagonalizable only if A has n different eigenvalues. (ii) If 0 is an eigenvalue of A then A is not singular. (iii) The system AX B, always have no solution if A-1 does not exist. (iv) If c is an eigenvalue of A then c 2 2c is an eigenvalue of A2 2 A . 9 d) Let S v1 , v2 , v1 2,1, v2 2,1 (i) S is a basis of R 2 . (ii) W= span S, has dimension = 1. (iii) If the vector u is not in span S then v1 , v2 , u is linearly independent set. (iv) The vector (0,0) is not in span S. 10