Tamil Nadu Open University (TNOU) 2008-3rd Year B.Sc Mathematics LINEAR ALGEBRA AND NUMBER SYSTEM UG–320 BMS–08 - Question Paper

 

 

B.Sc. DEGREE EXAMINATION
JUNE 2008.

(AY 20052006, CY 2006 batches only)

Third Year

Mathematics

LINEAR ALGEBRA AND NUMBER SYSTEM

Time : 3 hours Maximum marks : 75

PART A (5 5 = 25 marks)

Answer any FIVE questions.

Each question carries 5 marks.

1.         Define vector space over a field. Give an example.

2.         If is linearly independent in a vector space, prove that is also linearly independent.

3.         Compute the inverse of the matrix .

4.         Verify the Cayley-Hamilton theorem for the matrix .

5.         Find the rank of the matrix .

6.         Prove that the number of primes is infinite.

7.         Find the reminder when is divisible by
17 is 1.

8.         Prove that for any integer is divisible by 30.

PART B (5 10 = 50 marks)

Answer any FIVE questions.

Each question carries 10 marks.

9.         If is a non empty subset of a vector space over a field , prove that

             (a) is a subspace of

             (b)

             (c) is the smallest subspace of containing .

10.       Explain Gram-Schmidt orthogonalisation process. Using it find the orthonormal basis of with the basis .

11.       Define orthogonal complement of a subset of an inner product space and prove that it is a subspace.

12.       If and are two matrices prove that

             (a)

             (b) .

13.       (a) Prove that is orthogonal matrix.

             (b) Prove that .

14.       Find the eigen values and eigen vectors of

            

15.       State and prove Wilsons theorem.

16.       Prove that .