Indian Institute of Technology Kharagpur (IIT-K) 2009 M.Sc Mathematics 2nd year B.Tech (Linear Algebra) - Question Paper
INDIAN INSTITUTE OF TECHNOLOGY, Kharagpur
Departments: IM,NA and EC
MA20105 Linear Algebra
Autumn End semester, 2010 No. of students:100
Time:3 hrs Full Marks:50
INSTRUCTION: ans any 10 ques.. every ques. carries equal marks
Summary: This exam is conducted in the name of basic engineering to Undergraduate students of the departments Industrial,Ocean and naval architecture and also electronic , who entered IIT on the basis of JEE and persuing their B.tech or dual degree course in IIT Kharapgur
Subject mainly consists of vector spaces and their detail usages, Matrices and related things which will provide more understanding towards the subject
Indian Institute of Technology Kharagpur qtj Departments: IM, NA and EC.
MA20105 Linear Algebra Autumn End Semester Examination, 2010 No. of Students: 100
Full Marks: 50, Time: 3 Hrs.
INSTRUCTION: Answer any 10 questions. Each question carries equal marks.
1. (a) Is V = R2 a vector space over M with respect to the operations:
(a, b) + (c, d) = (a + c, b + d) and k(a, 6) = (fc2a, k2b)7
(b) Let V be a vector space of dimension n over a field F. Then show that any subset of n vectors of V that generates V is a basis of V.
(2+3 = 5 marks)
2. (a) Find the dimension of the vector space U of n x n symmetric matrices over a
field F?
(b) Let V be the vector space of all real polynomials P(x) and T : V V is defined by T(P{x)) = xP{z),P{x) 6 V, D : V -> V is defined by D(P(x)) = P(x), P(x) 6 V. Describe the mappings TD and DT. Are they equal?
(2+3 = 5 marks)
3. (a) Determine the linear mapping T : R3 E3 which maps the basis vectors
(0,1,1), (1,0, l)j (1,1,0) of R3 to (1,1,1), (1,1,1), (1,1,1) respectively. Verify that dimker(T) + dim Range(T) =3.
(b) Prove that each eigen value of a real orthogonal matrix has unit modulus.
(3+2 -- 5 marks)
P. T. 0
1
4. (a) Let U and V be vector spaces over the field F and let T be a linear transfor
mation from U into V. If T is one-one and onto, then show that the inverse function is a linear transformation from V into U.
(b) The set {I, t, e\ e{} is a basis of a vector space V of functions / : K -* 1. Let D be the different in the given basis.
Wf
D be the differential operator on V, that is, D(f) = Find the matrix of D
at
(2+3= 5 marks)
5. (a) Prove that two finite dimensional vector spaces V and W over a field F are
isomorphic if and only if dim V = dim W.
(b) Let V be an inner product space. Show that
(i) {&,?) = 0, V0 mV.
(ii) If (cx,/3) = 0, V/? in V, then a = 6.
(3+2= 5 marks)
6. State and prove the Cayley-Hamilton theorem for an nxn matrix A.
(5 marks)
7. (a) If Wi and W2 are subspaces of the vector space V(F), then prove that
(i) Wi + W2 is a subspace of V(F).
(ii) L(Wi U W2) = Wl + W2, where L{Wl U W2) is the linear span of [Wx U W2).
(b) If c e F is an eigen value of a linear operator T on a vector space V(F), then show that for any polynomial P(x) over F, P(c) is an eigen value of P{T).
(3+2= 5 marks)
8. Find whether the matrix 4= 3 2 1 j is diagonalizable. If so, find the matrix
V2 1 -V
P_1 and show that P 1AP D.
(5 marks) P. T. 0
9. (a) Prove that if a and (3 are vectors in. a Unitary space, then
(i) 4(a,0) = ||a + /?||2 - jja - /?||2 + i\\a + i/3\\2 - i\\a - i(3||2.
(ii) {a, 0) ~ Re (a, /?) + iRe (a, ip).
(b) A linear mapping T : R1 R3 is defined by T{x,y,z) = (x y,x + 2y>y + 3z), V(x, y, z) R3. Show that T is non-singular and determine T~l.
(3+2= 5 marks)
10. (a) State and prove the Cauchy-Schwarzs inequality for an inner product space V. (b) Give definition of an orthonormal set with an example.
(4+1= 5 marks)
11. (a) State the Gram-Schmidt orthogonalization process. If S = {a!i,a2,... ,Qm} is
m
an orthonormal set in V and if (3 6 V then show that 7 = 0 is
i=l
orthogonal to each of i, a2l..., am and, consequently to the subspace spanned by S'.
(b) Let V be the vector space of all functions from the real field R into IR. Let U be the subspace of even functions and W the subspace of odd functions. Show that V = U W.
(3+2= 5 marks)
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