LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

   B.Sc. DEGREE EXAMINATION – MATHEMATICS

QB 02

FIFTH SEMESTER – November 2008

MT 5502 – LINEAR ALGEBRA

 

 

 

Date : 10-11-08                     Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

SECTION – A

Answer all questions:                                                                        (10 x 2 = 20 marks)

 

1. Show that union of two subspaces of V need not be a subspace of V.
2. If the set {v₁, v₂, ….v_m} is a linearly independent set of vectors of a vector space V then prove that any non-empty set of this set is linearly independent.
3. Define basis of a vector space and give an example.
4. Show that  defined by  is a vector space homomorphism.
5. Define an inner product space and give an example.
6. Define an Eigen value and an Eigen vector.
7. If {v_i} is an orthonormal set, then prove that the vectors in {v_i} are linearly independent.
8. If A and B are Hermitian, Show that AB_BA is skew-Hermitian.
9. Let R³ be the inner product space over R under the standard inner product. Normalize .
10. Prove that the product of two invertible linear transforms on V is itself an invertible linear transformation on V.

SECTION – B

 

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

 

1. Show that } is a basis of the Vector space F[x] of all polynomials of degree at most n.
2. If A and B are subspaces of a vector space V over F, prove that (A+B) / B A / AB.
3. State and prove Schwarz’s inequality.
4. Prove that  is invertible if and only if the constant term of the minimal polynomial for T is not zero.
5. If dim V=n and , then prove that T can have atmost n distinct eigen values.
6. Let V=R³ and suppose that  is the matrix of  relative to the standard basis (1,0,0), (0,1,0), (0,0,1). Find the matrix relative to ,  & .
7. Let A be an mxn matrix over a field F, and let r be its rank. Then prove that is equal to the size of the largest non-singular square submatrix of A.
8. If  in V, then prove that T is unitary.

 

SECTION – C

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

 

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

W,  then prove that .

1. Find the Co-ordinate vector of (2, -1, 6) of R³ relative to the basis .                                                                   (15+5)
2. If then prove that is of dimension m².
3. a) State and prove Gram-Schmidt ortho normalisation theorem.
4. Normalize in C³, relative to the standard inner product. (15+5)
5. a) Prove that is invertible if and only if the constant term in the minimal polynomial for T is not zero.
6. b) 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. (14+6)

 

 

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