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MATHEMATICS PAPER III
LINEAR ALGEBRA and VECTOR CALCULUS
Linear Algebra
Unit - 1
Vector Spaces
Vector Spaces - General Properties
Vector Subspaces - Algebra
Linear Combinatons of vectors
Linear span
Linear sum of two subspaces
Linear dependence and Linear Independence of vectors
Basis of vector space
Finite dimensional vector spaces
Dimension of vector spaces, vector subspaces
Linear Transformations and Linear Operators
Null Space and Range of linear transformation
Rank and Nullity of linear transformation
Linear Transformations as vectors
Product of Linear Transformations
Invertible linear transformations
The Matrix Representation of a Linear Transformation
Unit - 2
The adjoint or transpose of a Linear Transformation
Sylvester's law of Nullity
Characteristic values and Characteristic vectors
Cayley-Hamilton Theorem
Diagonalizable Operators
Inner Product Spaces
Euclidean and unitary spaces
Norm or length of a vector
Schwartz Inequality
Othogonality
Orthonormal set, Complete Orthonormal set
The Gram-Schmidt Orthogonalization Process
(Linear Algebra by 1.J.N.Sharma and A.R.Vasista, Krishna Prakasham Mandir, Meerut-250002.)
(Linear Algebra by 2.Kenneth Hoffman and Ray Kunze, Pearson Education, New Delhi.)
(Linear Algebra by 3.Stephen H.Friedberg and others, published by Prentice-Hall International, Inc.)
Multiple Integrals and Vector Calculus
Unit - 3
Multiple Integrals
Introduction, the concept of a plane, curve
Line Integrals - Sufficient condition for the existence of the integral
The area of a subset of R²
Calculation of double integrals
Jordan curve, Area
Change of the order of the integration
Double integral as a limit
Change of a variable in double integration
Lengths of curves
Surface Areas
Integral expression for the length of a curve
A Course of Mathematical Analysis by Shanti Narayana and P.k.Mittal, S.Chand Pub.- Chapters 16 & 17.
Unit - 4
Vector Calculus
Vector Differentiation
Ordinary derivatives of vectors
Space curves
Continuity and Differentiability
Gradient
Divergence
Curl Operators
Formulae involving these operators
Vector Integration
Theorems of Gauss and Stokes
Greens theorem in plane
Aplications of these theorems
1.Vector Analysis by Murray. R.Spiegel, Schaum Series Pub. Com. Chapters - 3,4,5,6 & 7
2.Mathematical Analysis by S.C.Mallik and Savitha Arora, Wiley Eastern Ltd.
PAPER III PRACTICAL QUESTION BANK
Time : 3 hours Marks :
50
UNIT-I (LINEAR ALGEBRA-I)
1)Let V be the set of all pairs (x,y) of real numbers and let F be the field of real numbers.
Define: , + , = + ,0 &
, =
,0 .Is V with these operations a
vector space over the field of real numbers?
2)Is the set of all polynomials in x of degree ≤2 a vector space ? Justify.
3) Let R be the field of real numbers. Which of the following are subspaces of
(R)
(i){ ,2,3 :,, ∈ } (ii)
,, : ∈ }(iii)){ ,, :,, !"#} '
4) Which of the following sets of vectors $ = , , ,……. in are subspaces of
' % '
(n ≥ 3? (i) all$ s.t ≤ 0(ii) all$ s.t is an integer
(iii) all$ s.t + + +....+ =) ( k is a given constant).
% '
5) LetV= and W be the set of all ordered triads (x,y,z) such that − 3 + 4 = 0
Prove that W is a subspace of .
6) In the vector space ,determine whether or not the vector (3,9,-4,2) is a linear
combination
of the vectors (1,-2,0,3),(2,3,0,-1) and (2,-1,2,1). ,
7)Determine whether the vector (3,-1,0,-1) in the subspace of spanned by the vectors
(2,-1,3,2), (-1,1,1,-3) and (1,1,9,-5).
8)Find whether the following sets are linearly dependent or independent:
1 −2 −1 1
a) {(1,1,-1), (2,-3,5), (-2,1,4)} of b)-. 0,. 01of2 (R)
−1 4 2 −4 %3%
9) Determine whether the following vectors form basis of the given vector spaces
a) (2,1,0), (1,1,0), (4,2,0) of b) % + 3 − 2 , 2% + 5 − 3 , −% − 4 + 4of5
10) a) Show that the vectors (2,1,4), (1,-1,2), (3,1,-2) form the basis of %
b) Determine whether or not the vectors : (1,1,2), (1,2,5), (5,3,4) form a basis of .
11) Let V= and W be the subspace of given by 6 = { ,, : − 3 + 4 = 0}.
Prove
that Wis a subspace of and find its dimension.
12)Let F be the field of complex numbers and let T be function from 7 7 defined by
T( , , = − +2 , 2 + − , − −2 .Verify that T is linear
% % % %
transformation anddescribe the null space of T.
13) Show that the mapping 8:%−→ defined as T(a,b)=(a-b, b-a, -a) is a linear
transformation from % .Find the Range ,Rank,Nullspace and Nullity of T.
14) Let F be a subfield of complex numbers and let T from 7 7 defined by
T(a,b,c)= (a-b+2c, 2a+b, -a-2b+2c ) s.t T is a L.T find also the Rank and Nullity of T.
15) Let 8:−→ be the linear transformation defined by T(x,y,z)=( x+2y-z, y+z, x+y-2z
)
Find a basis and dimension of (i) the Range of T (ii) the Nullspace of T
16)Describe explicitly theL.T. 8:%−→ % such that T(2,3)=(4,5) and T(1,0)=(0,0).
17) Let 8 and 8 be two linear operators defined on
(R) by 8 (a,b,c)=(a+b, 2b, 2b-a )
%
8 (a,b,c)=(3a, a-b, 2a+b+c ) for all (a,b,c ) ∈
(R) show that 8 8 ≠8 8
% % %
18)Show that the operator T on defined by T(x,y,z)=(x+z, x-z, y ) is invertible and find
similar rule defining 8;.
19)Let T be the linear operator on defined by
T( , , ) = (3 + ,−2 + ,− +2 4 ) What is the matrix ofT in the ordered
% % %<
basis {$ ,$ ,$ } where $ =(1,0,1) $ =(-1,2,1) and $ =(2,1,1)?
% %
20) Find the matricesof the L.T. T on
defined as T(a,b,c)=(2b+c, a-4b, 3a) w.r.tstandard
>
ordered basis B={(1,0,0),(0,1,0),(0,0,1)},and orderedbasis= .={(1,1,1),(1,1,0),(1,0,0)}
UNIT-II(LINEAR ALGEBRA-II)
21) Find all (complex) proper values and proper vectors of the following matrices
0 1 1 1
a). 0b). 0
0 0 0
22) Let T be the linear operator on which is represented in the standard ordered basis by
the −9 4 4
matrix? D. Prove that T is diagonalizable.
−8 3 4
−16 8 7 1 1
23)a) Determine whether the matrix A= . 0similar over the field R to a diagonal
matrix? −1 1
Is A similar over the field C to a diagonal matrix?
1 2
b) P.T matrix A= . 0 is not diagonalizable over the field C.
0 1 0 0
24) Show that the characteristic equation of the complex matrix E = ?1 0 "D is
−%−"−
=0. 0 1
25) Find all the eigenvalues andeigen vectors of the matrix E = ?3 2 4D
2 0 2
4 2 3
26) Find eigenvalues andeigen vectors of the matrix E = ? 5 −6 −6D
−1 4 2
3 −6 −4
27) Show that the distinct eigenvectors of a matrix E corresponding to distinct eigen values
of E are linearly independent.
28) If $,F are vectors in an inner product space
7 and a,b∈F , then prove that:
‖ ‖% | |%‖ ‖% I | |%‖ ‖%
i) $ + "F = $ +" $,F +I" F,$ + " F
ii)$,F = ‖$ + F‖% − ‖$ − F‖%.
, ,
29) Prove that if $,F are vectors in an unitary space then
‖ ‖% ‖ ‖% ‖ ‖% ‖ ‖%
(i) 4 $, F = $+F − $−F + $+ F − $− F .
(ii) $, F = $,F + $, F
‖ ‖ ‖ ‖ ‖ ‖
30) If in an inner product space $ + F = $ + F , then prove that the vectors $,F are
Linearlydependent. Give an example to show that the converse of this statement is false.
31) If $ = , ,….., ,F = " " " ,…….." ) ∈
then prove that :
% ' % ' '
($, F = " + " + " +……..+ " . defines an inner product on
.
% % ' ' '
32) If $ = , , F = " ," ∈
. Define : ($,F = " − " − " +4 "
% % % % % % %
Show that all the postulates of an inner product hold good.
33)Let V(C)be the vector space of all continuous complex-valued functions on the unit
interval, IIIIII
0≤ ≤ 1.If f(t),g(t) ∈ V,let us define :JK ,L M = N K LP. Show that all
the O
postulates of an inner product hold good.
34) Determine whether the following define an inner product in
:
%
($, F = 2 + 5 given by $ = , , F = ,
% % % %
35) Apply Gram-Schmidt process to the vectors F = 1,0,1 ,F = 1,0,−1 F = 0,3,4
%
To obtain an orthonormal basis for
with the standard inner product.
36) Prove that the vectors $ P F in a real inner product space are orthogonal if and only if
‖$+F‖%=‖$‖%+‖F‖% .
37) Prove that two vectors $ P F in a complex inner product space are orthogonal if and
only ‖ ‖% ‖ ‖% ‖ ‖%
If $ + "F = $ + "F for all pairs of scalars a and b.
38) a) Find a vector of unit length which is orthogonal to the vector $ =(2,-1,6) of
with
respect to the standard inner product.
b) Find two mutually orthogonal vectors each of which is orthogonal to the vector:
$ =(4,2,3) of
with respect to the standard inner product.
39) Let V be a finite-dimensional inner product space and let { $ ,$ ,………$ } be an
orthonormal basis for V. Show that for any vectors $,F in % '
' IIIIIIII
V, $,F =∑ $,$ F,$ .
RS R R
40) Given the basis (2,0,1),(3,-1,5) and (0,4,2) for
, construct from it by the Gram-
Schmidt
process an orthonormal basis relative to the standard inner product.
UNIT-III (MULTIPLE INTEGRALS) % %
41)Evaluate the following integral: + PP over [ 0,;0," ]
∬3;X
42)Evaluate the following integral: PPover [0,1;0,1].
Y Y ∬3
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