LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

AB 26

M.Sc. DEGREE EXAMINATION – MATHEMATICS

FIRST SEMESTER – November 2008

    MT 1804 – LINEAR ALGEBRA

 

 

 

Date : 04-11-08                 Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

Answer ALL the questions.

 

1. a) i) Prove that the similar matrices have the same characteristic polynomial.

OR

1. ii) Let T be a linear operator on which is represented in the standard ordered basis by the matrix . Prove that T is not diagonalizable. (5)
2. b) i) State and prove Cayley-Hamilton theorem.

OR

1. ii) Let V be a finite dimensional vector space over F and T a linear operator on V.Then prove that T is diagonalizable if and only if the minimal polynomial for T has the form p=where are distinct elements of F. (15)

 

1. a) i) Let V be a finite dimensional vector space. Let be subspaces such that with for. Then prove that are independent subspaces.

OR

1. ii) Let W be an invariant subspace for T. Then prove that the minimal polynomial for divides the minimal polynomial for T.                   (5)
2. b) i) State and prove Primary Decomposition theorem.

OR

1. ii) If, then prove that there exist k linear operators on V such that
2. Each is a projection.
3. .

3.

4. The range of is.

iii) Prove that if  are k linear operators which satisfy conditions 1, 2 and 3 of the above and if let be range of then.                                                                                                                       (8+7)

• a) i) Let T be a linear operator on a vector space V and W a proper T-admissible subspace of V. Prove that W and Cyclic subspace Z(a;T) are independent.

OR

1. ii) Let T be a linear operator on which is represented in the standard ordered basis by the matrix . Prove that T has no cyclic vector. What is the T-cyclic subspace generated by the vector (1,-1, 3)?                                           (5)
2. b) i) Let a be any non-zero vector in V and let be the T-annihilator of . Prove the following statements:
3. The degree of is equal to the dimension of the cyclic subspace      Z(a;T).
4. If the degree of is k, then the vectors a, Ta, ,… form the   basis for Z(a;T).
5. If U is the linear operator on Z(a;T) induced by T, then the minimal polynomial for U is .

OR

 

 

 

 

1. ii) Let T be a linear operator on a finite dimensional vector space V and let

be a proper T-admissible subspace of V. Prove that there exist non-zero vectors in V with respective T-annihilators such that V=and divides, k=2, 3…r.                                    (15)

 

1. a) i) Define a positive matrix. Verify that the matrix is positive.

OR

1. ii) Let V be a complex vector space and f a form on V such that f () is real for every. Then prove that f is hermitian. (5)
2. b) i) Let F be the field of real numbers or complex numbers. Let A be an nxn matrix over F. Then prove that the function g defined by is a positive form on the space if and only if there exists an invertible nxn matrix P with entries in F such that.
3. ii) State and prove Principle Axis theorem. (6+9)

Or

iii) State and prove Spectral theorem and hence prove if, then for.                                                                             (15)

 

1. a) i) Define a bilinear form on a vector space over a field. Let m and n be positive integers and F a field. Let V be the vector space of all mxn matrices over F and A be a fixed mxm matrix over F. If, prove that is a bilinear form.

Or

1. ii) State and prove polarization identity for symmetric bilinear form f. (5)
2. b) i) Let V be a finite dimensional vector space over the field of complex numbers. Let f be a symmetric bilinear form on V which has rank r. Then prove that there is an ordered basis for V such that the matrix of f in the ordered basis B is diagonal and f () =1, j=1,…,r. Furthermore prove that the number of basis vectors for which =1 is independent of the choice of basis.

Or

1. ii) If f is a non-zero skew-symmetric bilinear form on a finite dimensional vector space V then prove that there exist a finite sequence of pairs of vectors,with the following properties:

1) f ()=1, j=1,2,,…,k.

2) f ()=f ()=f ()=0,ij.

3) If is the two dimensional subspace spanned by and, then V=where is orthogonal to all and  and the restriction of f to  is the zero form.                                                           (15)

 

 

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