# mathc1921 - BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE...

This preview shows pages 1–2. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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 ,...
View Full Document

{[ snackBarMessage ]}

### Page1 / 2

mathc1921 - BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online