LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
M.Sc. DEGREE EXAMINATION – MATHEMATICS
FIRST SEMESTER – NOV 2006
AA 18
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 TW divides the characteristic polynomial
for T. Also prove that the minimal polynomial for TW divides the minimal
polynomial for T.
b)
(5)
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 p  ( x  c1 )........( x  ck ) where c1 ,......, ck are
distinct elements of F.
(15)
II )a) Let V be a finite-dimensional vector space. Let W1 ,......,Wk be the subspaces of
V and let W  W1  ......  Wk . Then prove the following are equivalent.
i) W1 ,......,Wk are independent.
ii) For each j, 2  j  k , we have W j  (W1  .........  W j 1 ) = {0}.
[OR]
Let  be a non-zero vector in V and let p be the T-annihilator of  .Then
prove that
i) If the degree of p is k, then the vectors  , T  , T 2 ,......, T k 1 form a
basis for Z ( ; T ) .
ii) If U is the linear operator on Z ( ; T ) induced by T, then the minimal
polynomial for U is p .
(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 c1 ,......, ck are the distinct characteristic values of T, then
prove that there exist linear operators E1 ,......, Ek on V such that
(i) T  c1 E1  .........  ck Ek ;
(ii) I= E1  .........  Ek ;
(iii) Ei E j  0, i  j ;
(iv) Ei 2  Ei
(v) the range of Ei is the characteristic space for T associated with ci
Conversely, if there exist k distinct scalars c1 ,......, ck and k non-zero linear
operators E1 ,......, Ek which satisfy conditions (i),(ii) and (iii), then show that T
is diagonalizable, c1 ,......, ck 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 R3 which is represented in the standard basis by
 2 0 0


the matrix  0 2 0  . Find the minimal polynomial for T.
 0 0 -1 


(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]
 x  ci
If ei   

i  j  c j  ci

 , then prove that E j  e j T  .

(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 f  ,   is 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 1 , prove that S
is the spectrum of T’ and f(T)= Uf(T)U 1 .
V a)
(15)
Find all bilinear forms of F 2 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)
b) 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
2x2 matrix
 0 1
 -1 0  .


(15)
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