LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

XZ 20

FIFTH SEMESTER – APRIL 2008

MT 5502 – LINEAR ALGEBRA

 

 

 

Date : 03/05/2008                Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

SECTION – A                                  

Answer ALL the questions.                                                             (10 x 2 = 20 marks)

 

1. Illustrate by an example that union of two subspaces need not a subspace of a vector space.
2. Give an example of a linearly dependent set of vectors in R² over R.
3. “Every linearly independent set of vectors is a basis.” True or False. Justify.
4. Show that kernel of a homomorphism in a vector space is a subspace.
5. If V is an inner product space show that where a and b are scalars.
6. For T є A(v)define eigen value and eigen vector of T.
7. Find the trace of
8. Is the following matrix skew symmetric?
9. If T є A(v)is a Hermitian show that all its eigen values are real.
10. When do you say that a linear transformation T on V is unitary?

 

SECTION – B                                  

Answer any FIVE questions.                                               (5 x 8 = 40 marks)

 

1. Let V be a vector space of dimension n and  be libearly independent vectors in V. Show that there exists n-r new vectors in V such that is a basis of V.
2. If V is a vector space of finite dimension that is the direct sum of its subspace U and W show that
3. Verify that defined by T(a,b)=(a-b, b-a, -a) for a, b є R is a vector space homomorphism. Find the rank and nullity of T.
4. State and prove triangular inequality.
5. Show the T є A(V)is invertible if and only if the constant term of the minimal polynomial for T is not zero.
6. If V has dimension n andT є A(V), show that the rank of T is equal to the rank of the corresponding matrix m(T) in Fn.

 

 

1. Check the consistency of the following set of equations.

x₁+2x₂+x₃+11

4x₁+6x₂+5x₃=8

2x₁+2x₂+3x₃=19

1. If for all v є V, show that T is unitary.

SECTION – C

Answer any TWO questions.                                               (2 x 20 = 40 marks)

 

1. (a) Find whether the vectors (2, 1, 1, 1), (1, 3, 1, -2) and (1, 2, -1, 3) in R⁴ are linearly dependent or independent.

(b) If U and V are vector spaces of dimension m and n respectively over F, show that the vector space Hom(U,V) is of dimension mn.

1. a) Apply Gram Schmidt orthonormalization process to obtain an orthonormal basis for the subspace of R⁴ generated by the vectors. (1, 1, 0, 1), (1, -2, 0, 0),

(1, 0, -1, -2).

1. b) If are distinct eigen values of T є A(V) and if v₁, v₂, …, v_n are eigen vectors of T belonging respectively, show that v₁,v₂,…,v_n are linearly independent over F.

1. a) Let V be a vector space of dimension n over F and let T є A(V). If m₁(T) and m₂(T) are the matrices of T relative to two bases {v₁, v₂, …, v_n} and {w₁,w₂,…,w_n} of V respectively, show that there is an invertible matrix C in F_n such that m₂(T)=Cm₁(T)C^-1.

1. b) Show that any square matrix A can be expressed uniquely as the sum of a symmetric matrix and a skew-symmetric matrix.
2. c) If A and B are Hermitian show that AB+BA is Hermitian and AB-BA is skew Hermitian.

1. a) Find the rank of

1. b) If T є A(V)show that T* є A(V)and show that

(i)

(ii)  

(iii) (T)^* = T^*

(iv) (T^*)^* = T

 

 

 

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