Kannur University 2011 M.Sc Mathematics ( Syllabus) MODEL MAT1C02: Linear Algebra - Question Paper

PART A
1. If dim V?n and ????? is a linear operator such that kerT?T?V? then prove
that nis even.
2. obtain the transpose Tt of the linear operator T on R2 provided by T?x,y???x?y,x? .
3. Verify Cayley- Hamilton theorem for the linear operator T?a,b???2a,a?b? on??.
4. Prove that the T- cyclic subspace Z??;T? is 1 dimensional if and only if ? is a
characteristic vector for T .
5. Show that ,if Vis an inner product space with inner product ?|? , then ??|???0for all
? implies that ??0 .
PART B
6. a. Show that for 2 n- dimensional vector space V and W over a field F and linear
transformation T:V?W the subsequent are equivalent.
i. T is invertible.
ii. T is non- singular.
iii. T is onto.
b. If T is a linear operator on a 2 dimensional vector space V over F, which is
represented by
the matrix
a b
c d
? ?
? ?
? ?
, show that T2??a?d?T??ad?bc?I?0 .
7. a. If V and W are finite dimensional vector space over a fieldF, show that
i. The null space of Tt is the annihilator of the range of T .
ii. The range of Tt is the annihilator of the null space of T .
b. define the dual basis of B = ??1,0,?1?,?1,1,1?,?2,2,0?? of ?? .
8. a. Let T be a linear operator on a finite dimensional vector spaceV, with distinct
characteristic values one two , , ,k cc?? c and let i W be the null space of
, 1,2,3 , i T?cIi? ?? k .Prove that T is diagonalizable if and only if the
characteristic polynomial for T is ? ?1? ?2 ? ?
1 2
d d dk
k x?c x?c ? x?c .
b. obtain the characteristic polynomial and minimal polynomial for the linear operator A
on R3 determined by the matrix,
7 four 1
4 seven 1
4 four 4
A
? ??
?? ?? ? ?
??? ? ??
.
9. a. For any finite- dimensional vector space V over the field F and for a linear operator
Ton V
Prove that T is triangulable if and only if the minimal polynomial for T is a product of
linear polynomials over F .
b. Let T be the linear operator onR2, whose matrix with respect to the standard basis is
0 1
1 0
? ??
? ?
? ?
. obtain the subspaces of R2 invariant under T .
10. a. Prove that if Tis a diagonalizable linear operator on a finite dimensional vector space
V and one two , , ,k c c? c are distinct characteristic values of T , then there exist k
projections one two , , k E E ?E on V such that:
i. one two k I?E?E ???E
ii. one 1 two 2 k k T?cE?cE ???cE
iii. 0, i j EE ? i?j
b. obtain the Rational form of the real matrix
0 one 1
1 0 0
1 0 0
? ? ??
? ?
? ?
??? ??
.
11. a. Prove the Polarization Identity for a complex inner product space.
b. If Vis a finite dimensional vector space and T is a linear operator on V , show that
the cyclic subspace Z??;T? is 1 dimensional if and only if ? is a characteristic
vector of T .
PART C
UNIT- I
12. a. If fis a non- zero linear functional on the vector space V then prove that the null
space of fis a hyperspace in V. Conversely prove that every hyperspace in V is the
null space of a non- zero linear functional on V.
b. Let one two , , , ,r gf f? f be linear functionals on a vector space V with respective
Null spaces one two , , , , r N N N ? N . Then prove that g is a linear combination of
1 two , , ,r f f? fif and only if N contains the intersection one 2, r N ?N ??N.
13. Prove that the collection of all linear transformations ranging from 2 finite dimensional
vector spaces is a finite dimensional vector space
UNIT- II
14. a. State and prove Cayley- Hamilton theorem.
b. Verify that the linear operator on R3whose matrix with respect to the standard basis is
5 six 6
1 four 2
3 six 4
? ? ??
?? ? ? ?
?? ? ???
is diagonalizable.
15. a. If Vis finite dimensional vector space and T is a linear operator with minimal
polynomial as the product of linear factors ,then prove that for any proper invariant
subspace W of Vthere exist ??V with ??Wsuch that ?T?cI???Wfor a few
characteristic value c of Tand deduce that T is a triangulable operator.
b. obtain all the sub spaces of R2which are invariant under the linear operator
? ? ? ? one two one 1 two T x,x ? x,x?x .
UNIT- III
16.State and prove the Primary Decomposition Theorem.
17.a Show that an orthogonal set of non-zero vectors is linearly independent.
b. obtain an orthogonal basis ofR3using Gram- Schmidt orthogonalisation process.

Earning:   Approval pending.