Tamil Nadu Open University (TNOU) 2009-3rd Year B.Sc Mathematics " LINEAR ALGEBRA AND NUMBER SYSTEM " UG 471 BMS 08 - Question Paper

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UG-471    BMS-08

B.Sc. DEGREE EXAMINATION -JANUARY 2009.

Third Year (A.Y. 2005-06 and C.Y. 2006 batches only) Mathematics LINEAR ALGEBRA AND NUMBER SYSTEM

Time : 3 hours    Maximum marks : 75

PART A (5 x 5 = 25 marks)

Answer any FIVE questions.

1.    Prove that the set of complex numbers C is a vector space over the field R.

2.    Define inner product space. Give an example.

3. Compute the inverse of the matrix

4. Verify Cayley-Hamilton theorem for the matrix

^rii -i r 10 2.

_v3 -¹ 3_y

		'1 2	3N
5.	Find the rank of the matrix	2 3	4
		[ 2	2

6.    Find the smallest number with 18 divisors.

7.    Show that x⁵ - x is divisible by 30.

8.    Show that 7²" + 16n -1 = 0 (mod 64).

PART B (5 x 10 = 50 marks)

Answer any FIVE questions.

9.    Let V be a vector space over a field F. Let

S = {v₁, v₂, , v_n} czV. Prove that the following are equivalent

(a)    S is a basis for V.

(b)    S is a Maximal linear independent set

(c)    S is a minimal generating set.

10.    If V is a finite dimensional vector space over a field F and W is a subspace of V, prove that

V

dim = dim V - dim W.

W

11.    Apply Gram-Schmidt process to construct orthonormal basis from the basis {(1, 0, 1), (1, 3, 1), (3, 2, 1)}.

12.    Find the eigen values and eigen vectors of the

(2 2 n

1 3 1 1 2 2

13. Reduce the matrix

'2 -2 0 6 4 2 0 2 1 -1 0 3 v1 -2 1 2,

to its normal

form.

14.    Define Eulers (j> -function. Find the value </> (N), if N = p^a q^b r^c where p,q,r,--- are all primes and

a, b, c, are integers.

15.    State and prove Fermats theorem.

16.    Show that 28! + 233 = 0 (mod 899) .

3    UG-471

Attachment: tamil-nadu-open-university--tnou--2009-b-sc-mathematics-49075-1.pdf

Earning:   Approval pending.