mathc1921 - BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE...

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Unformatted text preview: BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE, PILANI II SEMESTER, 2004-2005 MATH C192: Mathematics II Comprehensive Examination Date: May 3, 2005 Time: 3 hrs. Day : Tuesday Max. Marks: 120 Note: Answer Part A and Part B on separate answer books and write Part A or Part B on the right top corner of the answer books. All Parts of each question must be done in sequence. PART A Q1. Show that W = { ( x 1 ,x 2 , ··· x n ) ∈ V n | α 1 x 1 + α 2 x 2 + ··· + α n x n = 0 } is a subspace of V n , where α i ’s are given constant. Find dim( W ). [7] Q2(a). If T is an idempotent transformation on V , then prove that R ( T ) = N ( I- T ) [8] Q2(b). Let A be a square matrix of order n having k distinct eigenvalues λ 1 ,λ 2 ,..,λ k . Let v i be an eigenvector corresponding to the eigenvalues λ i , i = 1 , 2 ,..,k . Then prove that the set { v 1 ,v 2 ,..,v k } is linearly independent. [10] Q3(a). Given S 1 = { (1 , 2 , 3) , (0 , 1 , 2) , (3 , 2 , 1) } and S 2 = { (1 ,- 2 , 3) , (- 1 , 1 ,...
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mathc1921 - BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE...

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