LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
M.Sc. DEGREE EXAMINATION - MATHEMATICS
FIRST SEMESTER – APRIL 2008
XZ 25
MT 1804 - LINEAR ALGEBRA
Date : 28/04/2008
Time : 1:00 - 4:00
Dept. No.
Answer ALL Questions.
I. a) i) Let T be the linear operator on
1
3

basis by the matrix  2
2
2
2

characteristic vector of T.
Max. : 100 Marks
R 3 which is represented in the standard ordered
-1 

-1  . Find a basis of R 3 , each vector of which is a
0 
Or
ii) Let T be a linear operator on a finite dimensional vector space V. Let
c1 , c2 ... ck be the distinct characteristic values of T and let Wi be the null space of
(T- ci I). If W= W1  W2  ...  Wk then prove that
dim W= dimW1  dimW2  ...  dimWk .
(5)
b)
i) State and prove Cayley-Hamilton theorem
Or
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)
II. a) i) Let V be a finite dimensional vector space. Let W1 ,W2 ,...Wk be subspaces of V
let W  W1  W2  ...Wk . Then prove that the following are equivalent :
1) W1 ,W2 ,...Wk are independent.
2) For each j, 2  j  k , W j  (W1  W2  ...  W j 1 ) = {0}
Or
iii) Let T be a linear operator on a finite dimensional vector space V and let
E1 , E2 ... Ek are linear operators on V such that 1) each Ei is a projection
2) Ei E j  0 if i  j 3) I  E1  E2  ...Ek and let Wi is the range of Ei . If each
(5)
Wi is invariant under T then prove that T Ei = Ei T, i=1,2,..k.
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
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 R2 which is represented in standard ordered basis
0
0
2


by the matrix  0
2
0  . Prove that T has no cyclic vector. What is the
0
0 -1 

T-cyclic subspace generated by the vector (1,-1,3)?
Or
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
ii) If T is a nilpotent operator on a vector space V of dimension n then prove that
characteristic polynomial for T is x n
(15)
IV. 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
ii) Let T be a linear operator on a complex finite dimensional inner product space
V. Then prove that T is self-adjoint if and only if T  is real for every  in V.



(5)
b) i) Let f be the form on R2 defined by f   x1 , y1  ,  x2 , y2   = x1 y1  x2 y2 .Find the
matrix of f with respect to the basis {(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 1 , 2 ,...,r  a basis
for the finite dimensional subspace W of V. Let M be the rxr matrix with
entries M jk  f ( k ,  j ) 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
W W ' ={0} if and only if M is invertible and when this is the case V=W+W ' .
(15)
V. a) i) Let F be a field. Find all bilinear forms on the space F 2 .
Or
ii) State and prove polarization identity for symmetric bilinear form f.
(5)
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   {1 ,  2 ,... n } for V such that the matrix of f in the
1, j=1,2,..r 
ordered basis B is diagonal and f (  i ,  j ) = 

0, j>r

Or
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, (1 , 1 ),( 2 ,  2 ),...( k ,  k ) with the following properties:
1) f (  j ,  j )=1, j=1,2,,…,k.
2) f (  i ,  j )=f (  i ,  j )=f (  i ,  j )=0, i  j.
3) If W j is the two dimensional subspace spanned by  j and  j , then
V= W1  W2  ...Wk  W0 where W0 is orthogonal to all  j and  j and the restriction
of f to W0 is the zero form.
(15)
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