LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034 M.Sc. DEGREE EXAMINATION – MATHEMATICS
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FIRST SEMESTER – NOV 2006
MT 1804 – LINEAR ALGEBRA
Date & Time : 26-10-2006/1.00-4.00 Dept. No. Max. : 100 Marks
Answer ALL Questions.
I ) a) Let T be a linear operator on an n-dimensional vector space V. Then prove that the characteristic and the minimal polynomials for T have the same roots, except for multiplicities.
[OR]
Let W be an invariant subspace for T. Then prove that the characteristic polynomial for the restriction operator divides the characteristic polynomial for T. Also prove that the minimal polynomial for divides the minimal polynomial for T. (5)
- b) State and prove Cayley-Hamilton theorem.
[OR]
Let V be a finite-dimensional vector space V over F and let T be a linear transform on V. Then prove that T is diagonalizable if and only if the minimal polynomial for T has the form where are distinct elements of F. (15)
II )a) Let V be a finite-dimensional vector space. Let be the subspaces of V and let . Then prove the following are equivalent.
- i) are independent.
- ii) For each we have = {0}.
[OR]
Let be a non-zero vector in V and let be the T-annihilator of .Then prove that
- i) If the degree ofis k, then the vectors form a basis for.
- ii) If U is the linear operator on induced by T, then the minimal polynomial for U is. (5)
- b) State and prove the primary decomposition theorem.
[OR]
Let T be a linear operator on a finite-dimensional vector space V over F. If T is diagonalizable and if are the distinct characteristic values of T, then prove that there exist linear operators on V such that
(i) T;
(ii) I=;
(iii);
(iv)
(v) the range of is the characteristic space for T associated with
Conversely, if there exist k distinct scalars and k non-zero linear operators which satisfy conditions (i),(ii) and (iii), then show that T is diagonalizable, are the distinct characteristic values of T, and conditions (iv) and (v) are satisfied . (15)
III a) Write a note on the Jordon form.
[OR]
Let T be a linear operator on which is represented in the standard basis by the matrix. Find the minimal polynomial for T. (5)
- b) State and prove cyclic decomposition theorem.
[OR]
State and prove generalized Cayley-Hamilton theorem. (15)
IV a) Prove that a form f is Hermitian if and only if the corresponding linear operator T is self adjoint.
[OR]
If , then prove that . (5)
- b) i) State and prove Principal Axis Theorem.
- ii) Let V be a complex vector space and f a form on V such that fis real for every .Then prove that f is Hermitian. (9+6)
[OR]
Let T be a diagonalizable normal operator with spectrum S on a finite-dimensional inner product space V .Suppose f is a function whose domain contains S. Then prove that f(T) is a diagonalizable normal operator with spectrum f(S) .If U is a unitary map of V onto V’ and T’=UTU, prove that S is the spectrum of T’ and f(T)= Uf(T)U . (15)
V a) Find all bilinear forms of F over F.
[OR]
Let f be a non-degenerate bilinear form on a finite-dimensional vector space V.
Then prove that the set of all linear operators on V which preserve f is a group under the operation of composition. (5)
- Let V be a finite-dimensional vector space V over a field of characteristic zero, and let f be a symmetric bilinear form on V. Then prove that there is an ordered basis for V in which f is represented by a diagonal matrix.
[OR]
Let V be an n-dimensional vector space over a sub field of the complex numbers, and let f be a skew-symmetric bilinear form on V. Then prove that the rank r of f is even, and if r = 2k, then there is an ordered basis for V in which the matrix of f is the direct sum of the (n-r) x (n-r) zero matrix and k copies of the 2×2 matrix
. (15)
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