2004

1) Let U and W be subspaces of R3 for which dim U=I, dim W=2 and U⊂W. Show that R3−U+W and U∩W=10}.

[10M]

2) Let {e1,e2,e3} be the standard basis of R3 and T be a linear transformation from R3 into R2 defined by T(e1)=(2,3)′,T(e2)=(1,2)′ and T(e3)=(−1,−4)′ (where ′ means transpose).
(i) What is T(1,−2,−1)?
(ii) What is the matrix of T with respect to the standard bases of R3 and R2?

[10M]

3) Find a linear map T:R2→R4 whose range is generated by (1,2,0,−4) and (2,0,−1,−3) Also find basis and the dimension of the
(i) range U of T. (ii) kernel W of T.

[10M]

4) Find the eigen values and their corresponding eigen vectors of the matrix (243212001). Is the matrix diagonalizable?

[10M]

5) For what values of a has the system of equations x+2y+z=1αx+4y+2z=24x−2y+2αz=−1

[10M]

6) Determine an orthogonal matrix which reduces the quadratic form

to a canonical form. Also, identify the surface represented by Q(x1,x2,x3)=7.

[10M]

2003

1) Let VP3(R) be the vector space of polynomial functions on reals of degree at most 3, Let D; V→V be the differention operator defined by D(a0+a1x+a2x2+a3x3)=a1+2a2x+3a3x2
(i) Show that D is a linear transformation
(ii) Find kemel and image of D.
(iii) What are dimensions of V1, ker D and image D4?
(iv) Give relation among them of (iii).

[10M]

2) Find the eigen values and the corresponding eigerivectors of A=(122−2).

[10M]

3) Show that the vectors (1,2,1), (1,0,−1) and (0,−3,2) form a basis for R(3).

[10M]

4) Determine non-singular matrices P and Q such that the matrix PAQ is in canonical form, where

Hence find the rank of A.

[10M]

5) Find the minimum polynomial of the matrix.
A=(122212221)

and use it to determine whether A is similar to a diagonal matrix.

[10M]

6) Show that the quadratic form 2x2−4xy+3xy+6y2+6yz+8z2 in three variables is positive definite.

[10M]

2002

1) Let S1,S2,…….Sk be subspaces of a vector space V(F). Show that the following statements are equivalent:
(i) S1+S2+…+Sk is a direct sum of V(F)
(ii) (S1+S2+…….+Si)∩S,n1={0}, for i=1,2,………,k−1
(iii) x1+x2+……+xk=0,x1∈S1,t=1,2
k⇒x1=0 for t=1,2,………,k
(iv) d(S,+S2+….+Sk)=d(S1)+d(S2)+….+d(Sk).

[10M]

2) Given A=[32121232] For which values of a does the vecter sequence {yn}−0 defined by yn=(1+αA+α2A2)yn−1n=1,2.

[10M]

3) Show that if λ is a non-zero eigen value of the non-singular n-square matrix A, then |A|λ is an eigen value of adjA.

Also given an example to prove that the eigen values of AB are not necessarily the product of cigen values of A and that of B.

[10M]

4) Given the liner transformation Y=[110231−235]X show that it is singular and the images of the linearly independent vectors

X1=[1,1,1]′
Xi=[2,1,2]′
Xi=[1,2,3]′
are linearly depchdent.

[10M]

5.(i) Calculate f(A)=eA−e−A for

[5M]

TBC

5.(ii) The n×n matrix A satisfies A^{4}=-1.6 A^{2}-0.641. Show that limm→∞Am exists and determine this limit.

[5M]

6) Reduce A=[136−114511543] to normal form N and compute the matrices P and Q such that AQ=N

[10M]

2001

1) Let V be the vector space of polynomials in x with real coefficients of degree almost 2. Let t1,t2,t3 be any 3 distinct real numbers. Define L1:V→R by L1(f)=f(t),i=1,2,3. Show that
(i) L4,L2,L3 are linear functional on V.
(ii) {L1,L2,L3} is a basis for the dual space V of V.

[10M]

2) In the notation of (a) above, find a basis B={P1,P2,P3} for V which is dual to {L1,L2,L3} and also express each P∈V in terms of elements of B.

[10M]

3) Let V be the vector space of polynomials in x with complex coefficients. Define T:V→V by (Tf)(x)=xf(x) and U:V→V by U[∑ni=0cix′]=∑n−1i=0ci+1x′.
(i) Find ker T
(ii) Show that U is linear
(iii) Show that UT=I and TV≠I,I= Identity on V.

[10M]

TBC

4) Let A=(100101010). Show that for every interger n≥3An=An+2+A2−I and hence find the matrix A8.

[10M]

5) Find the characteristic and minimal polynomials of A=(20012000−1) and determine whether A is diagonalisable.

[10M]

6) Let A=(112−121013), Find an invertible 3×3 matrix P such that P′AP=D,D is diagonal matrix. Find D also.

[10M]

2000

1) Let C(R) be the vector space of complex numbers over the field of real numbers. Under what conditions of the real numbers α,β,γ,δ in the set
S={α+iβ,γ+iδ},i=√(−1) do we have L(s)=C(R), where L(s) denotes the linear spans of S? Justify your answer.

[10M]

2) Let D:V(IR)→V(IR):f(x)→df(x)dx
T:V(IR)→V(IR):f(x)→xf(x)
be linear transformations on V (IR), the vector space of all polynomials in an indeterminate X with real coefficients.
Show that
(i) DT−TD=I the identity operator
(ii) (TD)2=TD2+TD

[10M]

3) Find A−2 when A=[1202−1000−1] by using Cayley-Hamilton theorem.

[10M]

4) Show that every square matrix can be expressed uniquely as A+iB where A, B are Hermitian.

[10M]

56) Reduce the quadratic form x2+4y2+9z2+t2−12yz+6zx4xy−2xt−6zt to canonical form and find its rank and signature.

[10M]