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
- a) (i) Prove that similar matrices have the same characteristic polynomial.
(OR) (5)
(ii) Let T be the linear operator on Â3 which is represented in the standard ordered basis by
the matrix . Find the characteristic polynomial of A.
- 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.
- 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.
- c is a characteristic value of T.
- The operator (T – cI) is singular.
- det (T – cI) = 0.
(OR) (5)
(ii) Let W be an invariant subspace for T. Then prove that the minimal polynomial for Tw
divides the minimal polynomial for T.
- 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,…,ck are the distinct characteristic values of T, then prove that there exist linear
operators E1,…,Ek on V such that
- T = c1E1 +…+ ckEk.
- I = Ej +…+ Ek.
- EiEj = 0,i≠j.
- Each Ei 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.
- b) (i) State and prove Cyclic Decomposition theorem.
(OR) (15)
(P.T.O.)
- 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.
- 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)
- 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.
- 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).
- 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)
- ii) Define the quadratic form q associated with a symmetric bilinear form f and prove that
- 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 ={b1, b2, … bn} for V such that the matrix of f in the ordered basis B is diagonal and .
(OR) (15)
- 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),… (ak, βk) with the
following properties.
- f (aj, βj) = 1 , j=1,2,,…,k.
- f (ai, aj)=f(βi, βj)=f(ai,βi)=0,i≠j.
- c) If Wj is the two dimensional subspace spanned by aj and βj, then V=W1 Å W2Å …Wk Å W0
where W0 is orthogonal to all aj and βj and the restriction of f to W0 is the zero form.