LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – MATHEMATICS

FIRST SEMESTER – NOVEMBER 2010

MT 1810/ 1804 – LINEAR ALGEBRA

 

 

 

Date : 30-10-2011             Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

                                  

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

 

(OR)                                                                                        (5)

 

(ii) Let T be the linear operator on ³ which is represented in the standard ordered basis by

the matrix .  Find the characteristic polynomial of A.

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

 

(OR)                                                                                           (15)

(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 are distinct elements of F.

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.

1. c is a characteristic value of T.
2. The operator (T – cI) is singular.
3. det (T – cI) = 0.

(OR)                                                                                        (5)

 

(ii) Let W be an invariant subspace for T.  Then prove that the minimal polynomial for T_w

divides the minimal polynomial for T.

 

1. b) (i) State and prove Primary Decomposition theorem.

(OR)                                                                                           (15)

 

(ii) Let T be a linear operator on a finite dimensional space V.  If T is diagonalizable and if

c₁,…,c_k  are the distinct characteristic values of T, then prove that there exist linear

operators E₁,…,E_k on V such that

1. T = c₁E₁ +…+ c_kE_k.
2. I = E_j +…+ E_k.
3. E_iE_j = 0,i≠j.
4. Each E_i is a projection

III. a)  (i) Let W be a proper T-admissible subspace of V. Prove that there exists a nonzero a in

V such that W Ç Z (a ; T) = {0}.

(OR)                                                                                             (5)

 

(ii) Define T_– annihilator, T_–-admissible, Projection of vector space V and Companion

matrix.

 

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

 

(OR)                                                                                           (15)

 

(P.T.O.)

1. ii) Let P be an m x m matrix with entries in the polynomial algebra F [x]. The following are

equivalent.

 

• P is invertible
• The determinant of P is a non-zero scalar polynomial.
• P is row-equivalent to the m x m identity matrix.
• P is a product of elementary matrices.

 

1. a) (i) Let V be a complex vector space and f be a form on V such that f (a,a) is real for

every a.  Then prove that f is Hermitian.                                                                                          (5)

(OR)

1. ii) 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.

 

1. b) (i) Let f be a form on a finite dimensional vector space V and let A be the matrix of f in an

ordered basis B. Then f is a positive form iff A = A^* and the principal minors of A are all

positive.

(OR)                                                                         (15)

(ii) Let V be a finite-dimensional inner product space and f a form on V.  Then show that there is a

unique linear operator T on V such that f(a,b) = (Ta½b) for all a, b in V, and the map f ®T is an

isomorphism of the space of forms onto L(V,V).

 

 

 

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 (a;b) = L₁ (a) L₂ (b) is bilinear.

 

(OR)                                                                            (5)

 

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

                                                                                                            

 

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

B ={b_1, b_2, … b_n} for V such that the matrix of f in the ordered basis B is diagonal and  .

(OR)                                                                                (15)

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, (a₁, β₁), (a₂, β₂),… (a_k, β_k) with the

following properties.

1. f (a_j, β_j) = 1 , j=1,2,,…,k.
2. f (a_i, a_j)=f(β_i, β_j­)=f(a_i,β_i)=0,i≠j.
3. c) If W_j is the two dimensional subspace spanned by a_j and β_j, then V=W₁ Å W₂Å …W_k Å W₀

where W₀ is orthogonal to all a_j and β_j and the restriction of f  to W₀ is the zero form.

 

 

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