LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

XZ 25

M.Sc. DEGREE EXAMINATION – MATHEMATICS

FIRST SEMESTER – APRIL 2008

    MT 1804 – LINEAR ALGEBRA

 

 

 

Date : 28/04/2008            Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

Answer ALL Questions.

1. a) i) Let T be the linear operator on which is represented in the standard ordered

basis by the matrix .  Find a basis of, each vector of which is a

characteristic vector of T.

Or

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

be the distinct characteristic values of T and let be the null space of

(T-I). If W=then prove that

dim W=.                                                 (5)

 

1. b) i) State and prove Cayley-Hamilton theorem

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 subspaces of V

let Then prove that the following are equivalent :

1)  are independent.

2) For each j,,= {0}

Or

iii)  Let T be a linear operator on a finite dimensional vector space V and let

are linear operators on V such that 1) each is a projection

2) 3) and let is the range of.  If each

is invariant under T then prove that T=T, i=1,2,..k.                      (5)

1. b) i) Let T be a linear operator on a finite dimensional vector space V.  Suppose that

the minimal polynomial for T decomposes over F into a product of linear

polynomials.  Then prove that there is a diagonalizable operator D on V and

nilpotent operator N on V such that 1)  T= D+ N

2)  DN=ND

Or

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

prove that T has a cyclic vector if only if the minimal and characteristic

polynomial for T are identical.                                                                     (15)

 

 

III. a) i)   Let T be a linear operator on which is represented in 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)?

Or

1. ii) If U is a linear operator on the finite dimensional vector space W and if U has a

cyclic vector then prove that there is an ordered basis for W in which U is

represented by the companion matrix of the minimal polynomial for U.     (5)

 

1. b) i)  State and prove Cyclic Decomposition theorem.

Or

1. ii) If T is a nilpotent operator on a vector space V of dimension n then prove that

characteristic polynomial for T is                                                          (15)

 

1. a) i) Let V be a finite dimensional complex inner product space and f a form on V.

Then prove that there is an orthonormal basis for V in which the matrix of f is

upper-triangular.

Or

1. ii) Let T be a linear operator on a complex finite dimensional inner product space
2. Then prove that T is self-adjoint if and only if is real for every  in V.                                                                                                                                                                                                                                            (5)

 

1. b) i) Let f be the form on defined by f=.Find the

matrix of f with respect to the basis {(1,-1),(1,1)}.

1. ii) State and prove the spectral theorem. (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 rxr matrix with

entries and W the set of all vectors in V such that

f ()=0 for all  W. Then prove that W is a subspace of V and

={0} if and only if M is invertible and when this is the case V=W+W.

(15)

1. a) i) Let F be a field. Find all bilinear forms on the space .

Or

 

1. ii) State and prove polarization identity for symmetric bilinear form f. (5)

 

1. 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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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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