Tamil Nadu Open University (TNOU) 2008 B.Sc Mathematics 3rd Year " Linear Algebra And Number System" 320 - Question Paper
UG-320 BMS-08
B.Sc. DEGREE EXAMINATION -JUNE 2008.
(AY 2005-2006, CY 2006 batches only)
Third Year
Mathematics
LINEAR ALGEBRA AND NUMBER SYSTEM
Time : 3 hours Maximum marks : 75
PART A (5 x 5 = 25 marks)
Answer any FIVE questions.
Each question carries 5 marks.
1. Define vector space over a field. Give an example.
2. If {, v, w} is linearly independent in a vector space, prove that {u + v, u - v, u - 2v + w} is also linearly independent.
3. Compute the inverse of the matrix ( 2 -1 1
-15 6 - 5
v 5 - 2 2 ,
4. Verify the Cayley-Hamilton theorem for the
. (1 2N
matrix I
14 3,
(2 3 4 3 1 2 222
5. Find the rank of the matrix
6. Prove that the number of primes is infinite.
7. Find the reminder when 21000 is divisible by 17 is 1.
8. Prove that for any integer n, n5 - n is divisible by 30.
PART B (5 x 10 = 50 marks)
Answer any FIVE questions.
Each question carries 10 marks.
9. If S is a non empty subset of a vector space V over a field F , prove that
(a) L (S) is a subspace of V
(b) S c L (S)
(c) L (S) is the smallest sub space of V containing S .
10. Explain Gram-Schmidt orthogonalisation process. Using it find the orthonormal basis of V3 (R) with the basis {2, -1, 0), (4, -1, 0), (4, 0, -1)}.
11. Define orthogonal complement of a subset S of an inner product space V and prove that it is a subspace.
12. If A and B are two m x n matrices prove that
(a) (at f = A
(b) (A + B)T = At + Bt .
f cos# sin#|
13. (a) Prove that A = is orthogonal
v-sin# cos# J
matrix.
(b) Prove that ||x + j|| < ||x|| + ||;y||.
14. Find the eigen values and eigen vectors of
( 3 10 5
- 2 - 3 - 4 v 3 5 7 j
15. State and prove Wilsons theorem.
16. Prove that 712 ! +1 = 0 (mod 719).
3 UG-320
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Attachment: |
| Earning: Approval pending. |
