LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – MATHEMATICS

ZA 39

FIRST SEMESTER – April 2009

MT 1810 / 1804 – LINEAR ALGEBRA

 

 

 

Date & Time: 17/04/2009 / 1:00 – 4:00       Dept. No.                                                         Max. : 100 Marks

 

 

 

Answer ALL the questions.

 

1. a) i) Let T be a linear operator on a finite dimensional space V and let c be a scalar. Prove that the following statements are equivalent.
2. c is a characteristic value of T.
3. The operator (T–cI) is singular.
4. det (T–cI) =0.

OR

1. ii) Let T be a linear operator on which is represented in the standard ordered basis by the matrix A=. Prove that T has no characteristic values in R. (5)
2. b) i) Let T be a linear operator on a finite dimensional vector space V. Prove that the minimal polynomial for T divides the characteristic polynomial for T.

OR

1. ii) Let V be a finite dimensional vector space over F and T be a linear operator on V then prove that T is triangulable if and only if the minimal polynomial for T is a product of linear polynomials over F. (15)

 

1. a) i) Let V be a finite dimensional vector space. Let be independent subspaces such that , then prove that for.

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) Let T be a linear operator on a finite dimensional space V. If T is diagonalizable and if are the distinct characteristic values of T, then prove that there exist linear operators on V such that

1..

2. .
3. .
4. Each is a projection
5. The range of is the characteristic space for T associated with.

iii) If there exist k distinct scalars and k non-zero linear operators which satisfy conditions 1,2 and 3, then prove that T is diagonalizable , are the distinct characteristic values of T and conditions 4 and 5 are satisfied also.                                                   (15)

 

• 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) If U is a linear operator on a finite dimensional space W, then prove that U has a cyclic vector if and only if there is some ordered basis for W in which U is represented by the companion matrix of the minimal polynomial for U. (5)

 

 

 

1. b) i) ) Let a be any non-zero vector in V and let be the T-annihilator of . Prove the following statements:
2. The degree of is equal to the dimension of the cyclic subspace      Z(a;T).
3. If the degree of is k, then the vectorsa, Ta, ,… form the   basis for Z(a;T).
4. 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. Then 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 the matrix of a form on a real or complex vector space with respect to any ordered basis . Let f be the form ondefined by Find the matrix of f with respect to a basis {(1,-1), (1, 1)}.

OR

1. ii) Let T be a linear operator on a complex finite dimensional inner product space V. Then prove that T is self-adjoint if and only if is real for every in V.                                                                             (5)
2. b) i) Let f be the form on a finite-dimensional complex inner product space V. Then prove that there is an orthonormal basis for V in which the matrix of f is upper-triangular.
3. ii) Prove that for every Hermitian form f on a finite-dimensional inner product space V, there is an orthonormal basis of V in which f is represented by a diagonal matrix with real entries.        (6+9)

OR

iii) Let f be a form on a real or complex vector space V and a basis for the finite dimensional subspace W of V. Let M be the r x r matrix with entries and Wthe set of all vectors  in V such that

f ()=0 for all in W. Then prove that Wis a subspace of V,={0} if and only if M is invertible and when this is the case V=W+W.                                                                                          (15)

1. a) i) Let V be a vector space over the field F. Define a bilinear form f on V and

prove that the function defined by f () =LLis bilinear.

OR

1. ii) Define the quadratic form q associated with a symmetric bilinear form f and prove that . (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 () =

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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