LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

FIFTH SEMESTER – APRIL 2011

MT 5508/MT 5502 – LINEAR ALGEBRA

 

 

 

Date : 20-04-2011              Dept. No.                                                  Max. : 100 Marks

Time : 1:00 – 4:00

 

SECTION – A

Answer ALL questions                                                                                         (10 X 2 = 20 Marks)

 

1. Define a vector space over a field F.
2. Prove that R is not a vector space over C.
3. Define the kernel of a linear transformation.
4. Prove that in V₃(R), the vectors (1, 2, 1), (2,1,0) and (1, -1, 2) are linearly independent.
5. Define an inner product space.
6. State the triangle inequality for inner product space.
7. Define an orthonormal set in an inner product space.
8. Prove that (A+B)^T = A^T + B^T where A and B are two m X n matrices.
9. Define an invertible matrix.
10. Define Hermitian and unitary linear transformations.

SECTION – B

 

Answer any FIVE questions                                                                                (5 X 8 = 40 Marks)

 

11. Prove that any set containing a linearly dependent set is also linearly dependent.
12. Let V be a vector space over a field F. Then prove that S = {v_1, v_2, . . ., v_n} is a basis for V if and only of every element of V can be expressed as a linear combination of elements of S.
13. Prove that T : R²→R² defined by T(a, b) = (a+b, a) is a vector space homomrphism.
14. Prove that T Є A(V) is invertible if and only of T maps V onto V.
15. Let T Є A(V) and λ Є F. Then prove that λ is an eigenvalue of T if and only if λI-T is singular.
16. Show that any square matrix can be expressed uniquely as the sum of a symmetric and a skew – symmetric matrix.
17. Show that the system of equations

X+2y+z=11

4x+6y+5z=8

2x+2y+3z=19 is inconsistent.

18. If TЄA(V) is Hermitian, then prove that all its eigen values are real.

SECTION – C

 

Answer any TWO questions                                                                         (2 X 20 = 40 marks)

 

19. a) If V is a vector space of finite dimension and is the direct sum of its subspaces U and

W, then prove that dim V = dim U + dim W.

 

1. b) If A and B are subspaces of a vector space V over F, prove that (A+B)/BA/A

 

(10 x 10)

20.    If U and V are vector spaces of dimensions m and n respectively over F, prove that

Hom (U,V) is of  dimension mn.

 

21. a) Apply the Gram – Schmidt orthonormalization process to the vectors (1,0,1), (1,3,1)

and (3,2,1) to obtain an orthonormal basis for R^3.

 

1. b) State and prove Bessel’s inequality.                                             (10 + 10)

 

22. a) Let V=R³ and suppose that is the matrix of T Є A(V) relative to the

standard basis V₁ = (1,0,0), V₂ = (0, 1, 0), V₃ = (0,0,1). Find the matrix of T relative to

the basis W₁ = (1,1,0),  W₂ =  (1,2,0), W₃ = (1,2,1).

 

1. b) Show that the linear transformation T on V is unitary if and only if it takes an

orthonormal basis of V onto an orthonormal basis of V.                                     (10 + 10)

 

 

 

Go To Main Page

Latest Govt Job & Exam Updates:

View Full List ...