Post on 26-Mar-2018
Vector Calculus & Linear Algebra
Tutorial Book Year - 2012
GENERAL DEPARTMENT
L. E. COLLEGE, MORBI
My
Beloved Students
With all warm regards and wishes I am glad to present you this fruit of toil taken by the Professors of General Department. This manual is designed in such a way that it becomes useful in grooming you in a better way. It applies the concept that you study in your theory classes.
I hope this labour will inculcate in you the practical wisdom which you require in your professional life. This will widen your horizon and deepen your knowledge for the subject.
This is the toil taken for you by your professors keeping in mind your need as a student. They have tried their level best to form a uniform manual which is perhaps the first in Degree side. I am glad to have such a team of intellectuals who worked hard and converted the idea into reality. I congratulate them all. I feel proud that L. E. College, Morbi is the pioneer in generating manual for Degree students in General side.
I wish you the very success in your life and pray to Almighty to help us to groom you into a better Engineer……..
Prof. P.C.Vasani
Principal,
L. E. College, MORBI
ॐ સહના વવત ુસહનૌભનુક્ત ુસહવીર્યમ કરવા વહ ે
તેજસ્વીના વદી તમસ્ત ુમાાં વવદ્ વવસા વહ ે
ॐ શાાંવત શાાંવત શાાંવત
Acknowledgements
We heartily extend our vote of thanks to the Principal. Prof. P.C.Vasani, L.E.College,morbi to guide us and permit us to bring our vision into a reality. We are also grateful to our Head of the Department, Prof.Y.N. Dangar for his constant support and encouraging attitude.
Our special thanks are due to our entire staff member who supported us in compiling our work. Last but not the least to Almighty for his blessings. Compiled by:
Dr. K.K.Kanani
Asst. prof. D.D.Pandya
Asst. prof. S.G.Sonchhabda
1
Certificate
This is to Certify that Shri________________________
Enroll. No.____________________ of B.E. _________________
Class has Satisfactorily Completed the Course in Vector Calculus
& Linear Algebra (110015) Tutorials within Four Walls of
LUKHDHIRJI ENGINEERING COLLEGE, MORBI.
Date of Submission_________ Staff in-charge_______________
Head of Department____________________________________
2
INDEX
Sr. No.
Name of Experiment Page No.
Date of Exp.
Performed Signature
1. Tutorial – 1 03
2. Tutorial – 2 04
3. Tutorial – 3 11
4. Tutorial – 4 17
5. Tutorial – 5 21
6. Tutorial – 6 28
7. Tutorial – 7 32
8. Tutorial – 8 41
9. Tutorial – 9 50
10. Tutorial – 10 55
11. Tutorial – 11 60
12. Tutorial – 12 66
13. Tutorial – 13 71
14. Tutorial – 14 75
15. Tutorial – 15 79
16. Tutorial – 16 85
3
Tutorial – 1
DEFINITION OF MATRICES (UNIT 6,7)
DEFINITIONS OF ALL TYPES OF MATRICES
4
5
TUTORIAL – 2 ALGEBRA OF MATRICES (UNIT 6,7)
Q-1 If 1 2
0 3A
and 3( ) 2 3 4p x x x then find out p(A).
Q-2 If 1 2
1 3A
then find out 3A and 3A .
Q-3 Find the inverse of given matrices by Gauss-Jordan method if it exists.
(1)
3 4 6
6 7 8
9 10 11
(2)
1 3 3
1 4 3
1 3 4
(3)
1 1 2
3 1 1
1 3 4
6
Q-4 Find the inverse of given matrices by Determinant method if it exists
7
(1)
1 0 1
1 1 1
0 1 0
(2)
3 3 4
2 3 4
0 1 1
(3)
1/ 3 2 / 3 2 / 3
2 / 3 1/ 3 2 / 3
2 / 3 2 / 3 1/ 3
8
Q-5 Find the Row Echelon form of the following.
(1)
3 1 1 2
1 1 1 2
2 2 1 6
(2)
1 8 10 5 3
0 1 13 9 12
0 0 0 1 1
0 0 0 0 0
Q-6 Find the Reduced Row Echelon form of the following.
(1)
1 3 2 0 1 0
0 2 1 1 0 1
0 0 0 2 1 0
0 0 0 0 0 1
(2)
1 9 0 0 2
0 0 1 0 16
0 0 0 1 1
0 0 0 0 0
9
Q-7 Solve the following system of equations by using Gauss Elimination method.
(1) 1 23 4 10x x & 1 25 8 17x x & 1 23 12 12x x
(2) 2 2 2 0, 2 5 2 1,8 4 1x y z x y z x y z Q-8 Solve the following system of equations by using Gauss Jordan Method.
(1) 1 2 3 4 57 2 2 4 3 8x x x x x & 1 2 4 53 3 2 1x x x x &
1 2 3 54 8 20 1x x x x
10
(2) 3 2 5x y & 2 2x z & 4 3 8y z .
11
**Ans**
========================================================
Q-(1) 9 2
0 13
Q-(2) 3A =11 30
15 41
& 3A = 41 30
15 11
Q-(3) (1)
1 16 / 3 10 / 3
2 7 4
1 2 1
(2)
7 3 3
1 1 0
1 0 1
(3)
7 2 31
13 2 710
8 2 2
Q-4 (1) no inverse (2)
1 1 0
2 3 4
2 3 3
(3)
1 2 21
2 1 23
2 2 1
Q-5 (1)
1 1 1 2
0 4 4 4
0 0 3 6
(2) already in row echelon form
Q-6 (1)
1 0 1/ 2 0 1/ 4 0
0 1 1/ 2 0 1/ 4 0
0 0 0 1 1/ 2 0
0 0 0 0 0 1
(2) Already in form
Q-7 (1) 1 23& 1/ 4x x
(2) ( 1/ 7) (3/ 7) , (1/ 7) (4/ 7) , ;x t y t z t t=any constant
Q-8 (1) 1 2 3 4 5x =(28/15)+(2/3)t+(1/3)s,x =-23/15,x =1+(1/3)t+(8/3)s,x =t,x =s
(2) No solution
12
TUTORIAL – 3
SOLUTION OF EQUATION USING MATRICES (UNIT 6,7) Q-1 Solve the following system of equations using Gaussian elimination method or
indicate nonexistence of solutions.
(i) x+2y-z=2, x-y+z=4, 2x+y-z=5 (ii) 2x+y-z=2, x-3y+z=1, 2x+y-2z=6 (iii) 10x+4y-2z=-4, -3w-17x+y+2z=2, w+x+y=6, 8w-32x+16y-10z=4
Q-2 Check whether the following system of equations has solutions or not.
(i) 1 2 3 53 2 0x x x x , 1 2 3 4 1x x x x , 2 3 4 54 2 4 3 3x x x x
1 2 3 43 2 2 0x x x x
(ii) 1 2 1 2 4 2 4 1 2 43 0 & 1& 4 4 2 & 3 2 0x x x x x x x x x x
13
Q-3 Under which condition the system of equations is consistent.
1 2 3 53 2x x x x a , 1 2 3 4x x x x b , 2 3 4 54 2 4 3x x x x c
1 2 3 43 2 2x x x x d
Q-4 For what choices of parameters are the following system consistent?
(i) x+y=a, 2x+2y=b, 3x+3y=c, 4x+4y=d (ii) x+y-z=1, x-az=-1, x+ay=-1
14
Q-5 For what choices of parameters do the following systems have a unique
solution?
(i) x+y-z=-2, x-az=-1, x+ay=-1
(ii) v+w=a, u+aw=1, u+v=0, av-w=1 Q-6 Find the rank of the following matrices either by method of evaluating
determinants or by elementary transformations.
(i)
1 2 3
2 4 7
3 6 10
(ii)
3 0 2 2
6 42 24 54
21 21 0 15
(iii)
1 3 5 7
3 5 7 9
5 7 9 11
7 9 11 13
(iv)
1 2 1
1 2 2
1 2 3
1 2 4
(v)
1 3 1 2
0 11 5 3
2 5 3 1
4 1 1 5
(vi)
1 2 3 2
2 5 1 2
3 8 5 2
5 12 1 6
15
Q-7 Show that two planes x+2y+3z=1 and x+2y+3z=2 have no points of
intersections.
16
Q-8 Show that there is no line in 2R containing the points (1,1), (3,5), (-1,6) and (7,2)
17
@@ANS@@
Q-1 (i) x=3,y=0,z=1
(ii) x=-1/7, y=-12/7, z=-4 (iii) w=4, x=0, y=2, z=6
Q-2 (i) system has no solutions (ii) system has no solutions Q-3 -3a-2b+c+d=0 Q-4 (i)Consistent if b=2a,c=3a,d=4a
(ii)Consistent if a=0,a 0,2
Q-5 (i)system is consistent if a 2.Unique solution exists if a 0 or 2
(ii)system is consistent if a=-1 or 2. Unique solution exists if a=2.
Q-6 (i)2 (ii)2 (iii)2 (iv)2 (v)2 (vi)2
18
Tutorial – 4
RANK OF MATRICES (UNIT 6,7) Q-1 Reduce the following matrices to normal form and find its rank
(1)
1 1 2 3
4 1 0 2
0 3 1 4
0 1 0 2
(2)
1 1 3 6
1 3 3 4
5 3 3 11
(3)
0 1 3 1
0 0 1 1
3 1 0 2
1 1 2 0
(4)
2 3 2 5 1
3 1 2 0 4
4 5 6 5 7
(5)
1 3 2 5 1
2 2 1 6 3
1 1 2 3 1
0 2 5 2 3
19
Q-2 Find nonsingular matrices P and Q such that PAQ is in the normal form. Hece find rank of matrix.
(1) 1 2 3
3 1 2
(2)
5 3 14 4
0 1 2 1
1 1 2 0
(3)
1 1 2
1 2 3
0 1 1
(4)
2 1 3 6
3 3 1 2
1 1 1 2
(5)
1 2 2 1
4 2 1 2
2 2 2 0
20
21
ANSWERS
Q-1 (1)4 (2)3 (3)2 (4)2 (5)3 Q-2 (1)2 (2)3 (3)2 (4)3 (5)3
22
Tutorial – 5
SOLUTION OF EQUATION USING MATRICES (UNIT 6,7) Q-1 Using Cramer’s rule solve the following system of linear equations.
(1) 2 6, 3 4 6 30, 2 3 8x z x y z x y z
(2) 4 6,4 2 1,2 2 3 20x y z x y z x y z
(3) 3x-y+z=4,-x+7y-2z=1,2x+6y-z=5
Q-2 Find the solutions of the following system of linear equations using 1A .
(1) 2 3 5,2 5 3 3, 8 17x y z x y z x z
(2) x+y+z=6, 2x-3y+4z=8, x-y+2z=5
23
Q-3 Investigate for what values of and the following system of equations have
(a)no solution (b) a unique solution (c) an infinite number of solutions
(1) x+y+z=6, z+2y+3z=10, x+2y+ z=
(2) 2x+3y+5z=9, 7x+3y-2z=8, 2x+3y+ z=
(3) x+2y+3z=6, x+3y+5z=9,2x+5y+ z=
24
Q-4 Determine the values for which the equations 22 3, ,3 3x y z x y z x y z are consistent and solve them for these
values of .
Q-5 Show that the equations 4 5 ,4 5 6 ,5 6 7x y z a x y z b x y z c do not
have a solution unless a+c=2b
Q-6 Show that the system of equations
2 3 ,3 2 ,2 3x y z x x y z y x y z z can possess a non-trivial solution
only if =6.
25
Q-7 Find the value of k, such that the system of equations
2 3 2 0,3 3 0,7 0x y z x y z x ky z has a non-trivial solution. Find the
solutions
Q-8 Show that the system of equations 0, 0, 0ax by cz bx cy az cx zy bz
has a non-trivial solution only if a+b+c=0 or a=b=c.
Q-9 Find the necessary and sufficient conditions on a,b and c for the system
3 & 2 &3 7x y z a x y b x y z c to be consistent. Also discuss about its
consistency when a=1,b=1,c=3 and a=1,b=0, c=-1.
26
Q-10 What conditions must 1 2 3,k k andk satisfy in order for the following system of
equations to be consistent?
(1) 1 2 32 , ,2 3x y z k x z k x y z k
(2) 1 2 3x+2y+3z=k ,2x+5y+3z=k ,x+8z=k
Q-11 Express the following matrix as the sum of a symmetric and skew-symmetric
matrices.
2 3 0
1 2 2
3 1 1
27
Q-12 Find l,m,n so that the given matrix is orthogonal.
0 2m n
l m n
l m n
Q-13 Is the given matrix is orthogonal? If not, can it be converted into an orthogonal
matrix.
2 2 1
2 1 2
1 2 2
Q-14 Show that the given matrix is nilpotent of index 2.
1 3 4
1 3 4
1 3 4
28
Answers
Q-1 (1) x=-10/11, y=18/11 z=38/11 (2) x=-144/55, y=-61/55, z=46/11
(3) Cramer’s rule does not apply
Q-2 (1) x=1,y=-1,z=2 (2) x=1,y=2,z=3
Q-3 (1) (a) =3, 0 (b) 3, R (c) =3, =10
(2) (a) =5, 9 (b) 5, R (c) =5, =9
(3) (a) 8, 15 (b) 8, R (c) 8, 15
Q-4 2,3 2( , 1, ) & 3( , 0, 3 )for x k y z k for x k y z k
Q-7 k=5,x=-7k/11,y=12k/11,z=k
Q-9 c-a-2b=0,consistent if a=1,b=1,c=3 and inconsistent if a=1,b=0,c=-1
Q-10 (1) 3 1 2k k k (2) no condition
Q-11
2 1 3/ 2 0 2 3/ 2
1 2 1/ 2 & 2 0 3/ 2
3/ 2 1/ 2 1 3/ 2 3/ 2 0
B C
Q-12 1 1 1
, ,2 6 3
l m n
Q-13 no but if we take B=1/3A then it is orthogonal matrix.
29
Tutorial – 6 vectors in nR (UNIT 5)
Q-1 In each part, compute the Euclidean norm of the vectors.
(a) (-2,5) (b) (1,2,-2) (c) (3,4,0,-12) (d) (-2,1,1,-3,4)
Q-2 Show that if v is a nonzero vectors in nR ,then (1/||v||)v has Euclidean norm 1.
Q-3 Let v=(-2,3,0,6). Find all scalars k such that ||kv||=5
Q-4 Find the Euclidean inner product u v
(a) u=(2,5),v=(-4,3) (b) u=(4,8,2),v=(0,1,3)
(c) u=(3,1,4,-5),v=(2,2,-4,-3) (d) u=(-1,1,0,4,-3),v=(-2,-2,0,2,-1)
Q-5 Find two vectors in 2R with Euclidean norm 1 whose Euclidean inner
product with (3,-1) is zero.
Q-6 Find the Euclidean distance between u and v.
30
(a) u=(1,-2),v=(2,1) (b) u=(2,-2,2),v=(0,4,-2)
(c) u=(0,-2,-1,1),v=(-3,2,4,4) (d) u=(3,-3,-2,0,-3),v=(-4,1,-1,5,0)
Q-7 In each part, determine whether the given vectors are orthogonal.
(a) u=(-1,3,2),v=(4,2,-1) (b) u=(-2,-2,-2),v=(1,1,1)
(c) u=(u,v,w),v=(0,0,0) (d) u=(-4,6,-10,1),v=(2,1,-2,9)
(e) u=(0,3,-2,1),v=(5,2,-1,0) (f) u=(a,b),v=(-b,a)
Q-8 For which values of k are u and v orthogonal?
(a) u=(2,1,3), v=(1,7,k) (b) u=(k,k,1), v=(k,5,6)
Q-9 In each part, verity that Cauchy-Schwarz inequality holds.
(a) u=(3,2), v=(4,-1) (b) u=(-3,1,0), v=(2,-1,3)
(c) u=(-4,2,1), v=(8,-4,-2) (d) u=(0,-2,2,1), v=(-1,-1,1,1)
Q-9 Find u v given that ||u+v||=1 and ||u-v||=5
31
Q-10 Use the Cauchy-Schwarz inequality to prove that for all real values of a,b and .
2 2 2( cos sin )a b a b
Q-11 Find the angle between a diagonal of a cube and one of its edges.
32
ANSWERS
Q-8 k=5
7 Q-10(a)
1 3 1 3, , ,
10 10 10 10
Q-13(a) k=-3 (b)k=-2,-3
Q-19 4/15( 1,1,2,3) &1/15(34,11,52, 27) Q-20 NO Q-251cos (1/ 3)
Q-26 expression is not unique ,no Q-27 (0, 1/ 2,1/ 2)u
33
Tutorial – 7 vector spaces (UNIT 5)
Q-1 Show that the set of polynomials with real coefficients defined by
2
0 1 2 0 1( ) { ( ) .... / & , ,...... }n
n nPn x p x a a x a x a x n a a a makes a
vector space when given the natural’+’ as
0 1 0 1( ) ( ) ( ... ) ( .... )n n
n np x q x a a x a x b b x b x =
0 0 1 1( ) ( ) ...( ) n
n na b a b x a b x and ‘.’ As
0 1 0 1( ) ( ... ) .....n n
n ncp x c a a x a x ca ca x ca x
34
Q-2 Show that the set V of all 22 matrices with real entries is a vector space under
standard addition and multiplication.
35
Q-3 Let 2V R with 1 2 1 2( , ) & ( , )u u u v v v then 1 1 2 2( , )u v u v u v and for scalar
k 1( ,0)ku ku does not form a vector space.
36
Q-4 Every plane through the Origin is a Vector Space.(Hint: Plane V passes through origin
form equation ax+by+cz=0 & if 1 2 3 1 2 3( , , ) & ( , , )u u u u v v v v are points in V then
1 2 3 1 2 30 & 0au bu au av bv cv .
37
Q-5 Let V is the set of positive real numbers with addition and scalar multiplication
defined by & cx y xy cx x . Show that this set V is a vector space.
Q-6 Let V be the points on a line through the origin in 2R with the standard addition and
scalar multiplication. Show that V is vector space.
38
((hint: (( , ) _( 0))x y Vthen ax by
Q-7 Let V be the points on a line that does not pass thorough the origin in 2R with the
standard addition and scalar multiplication. Show that V is not vector space.
((hint: (( , ) _( ))x y Vthen ax by c
39
Q-8 Show that for 2V R defined by 1 1 1 2 1 1 2 2( , ) ( , ) ( 2 , )u v v v u v u v is not vector
space.
40
Q-9 Show that
(a) The set of vectors of the form(a,0,0) is a subspace of 3R
(b) The set of vectors of the form(a,1,0) is not a subspace of 3R
(c) The set of vectors of the form(a,2a,3a) is a subspace of 3R
(d) 2 2 2{( , , ) | 1}x y z x y z is not a subspace of 3R
Q-10 (a) Let W be the set of matrices of the form
12
21 22
31 32
0 a
a a
a a
. Is this a subspace of
3,2M .
41
(b) Let W be the set of matrices of the form 12
22
2
0
a
a
. Is this a subspace of 2,2M
42
Tutorial – 8 LINEAR INDEPENDENT/LINEAR COMBINATION/BASIS (UNIT 5)
Q-1 which of the following are linear combinations of u=(0,-2,2) and v=(1,3,-1)?
(a) (2,2,2) (b) (3,1,5) (c) (0,4,5) (d) (0,0,0)
Q-2 Express the following as linear combinations of u=(2,1,4) &v=(1,-1,3) & w=(3,2,5).
(a) (-9,-7,-15) (b) (6,11,6) (c) (0,0,0) (d) (7,8,9)
43
Q-3 Express the following as linear combinations of 2
1p =2+x+4x , 2
2p =1-x+3x and 2
3p =3+2x+5x .
2 2 2(a) -9-7x-15x (b) 6+11x+6x (c) 0 (d) 7+8x+9x
Q-4 Which of the following are linear combinations of
4 0 1 1 0 2A= , ,
-2 -2 2 3 1 4B C
6 8
( )1 8
a
0 0
( )0 0
b
6 0
( )3 8
c
1 5
( )7 1
d
44
Q-5 IN each part, determine whether the given vectors span 3R .
(a) 1V =(2,2,2) , 2V =(0,0,3), 3V =(0,1,1)
(b) 1V =(2,-1,3) , 2V =(4,1,2) , 3V =(8,-1,8)
(c) 1V =(3,1,4) , 2V =(2,-3,5), 3V =(5,-2,9) , 4V =(1,4,-1)
(d) 1V =(1,2,6) , 2V =(3,4,1) , 3V =(4,3,1) , 4V =(3,3,1)
Q-6 Let f=cos2x and g=sin2x. Which of the following lie in the space spanned by f
And g?
(a) cos2x (b) 3+x2 (c) 1 (d) sinx (e) 0
45
Q-7 Determine whether the following polynomials span 2 p .
2 2 2
1 2 3 4P =1-x+2x , p =3+x , p =5-x+4x , p =-2-2x+2x
Q-8 Let u=(2,1,0,3), v=(3,-1,5,2) w=(-1,0,2,1).which of the following vectors are in
span{u,v,w}?
(1) (2,3,-7,3) (2) (0,0,0,0) (3) (1,1,1,1) (4) (-4,6,-13,4)
Q-9 Find equation for the plane spanned by the vectors u=(-1,1,1) and v=(3,4,4)
46
Q-10 Find parametric equation for the line spanned by the vector u=(3,-2,5)
Q-11 Which of the following set of vectors in 3R are linearly dependent?
(a) (4,-1,2),(-4,10,2) (b) (-3,0,4), (5,-1,2), (1,1,3)
(c) (8,-1,3), (4,0,1) (d) (-2,0,1), (3,2,5), (6,-1,1), (7,0,-2)
Q-12 Which of the following sets of vectors in 4R are linearly dependent?
(a) (3,8,7,-3), (1,5,3,-1), (2,-1,2,6), (1,4,0,3)
(b) (0,0,2,2),(3,3,0,0), (1,1,0,-1)
(d) (0,3,-3,-6), (-2,0,0,-6), (0,-4,-2,-2), (0,-8,4,-4)
47
Q-13 Which of the following sets of vectors in 2P are linearly independent?
2 2 2
2 2 2
2 2 2
2
(a) 2-x+4x , 3+6x+2x , 2+10x-4x
(b) 3+x+x , 2-x+5x , 4-3x
(c) 1+3x+3x , x+4x , 5+6x+3x , 7+2x-x
48
Q-14 Which of the following sets of vectors are bases for 2R ?
(a) (2,1), (3,0) (b) (4,1), (-7,-8)
(c) (0,0), (1,3) (d) (3,9), (-4,-12)
Q-15 Which of the following sets of vectors are bases for 3R ?
(a) (1,0,0), (2,2,0), (3,3,3) (b) (3,1,-4), (2,5,6), (1,4,8)
(c) (2,-3,1), (4,1,1), (0,-7,1) (d) (1,6,4), (2,4,-1), (-1,2,5)
49
Q-16 Which of the following sets of vectors are bases for 2P ?
2 2
2 2 2
(a) 1-3x+2x , 1+x+4x , 1-7x
(b) 4+6x+x , -1+4x+2x , 5+2x-x
Q-17 Show that the following set of vectors is a basis of 22M
3 6 0 1 0 8 1 0
, , ,3 6 1 0 12 4 1 2
50
51
Tutorial – 9 eigen value&eigen vector (UNIT 10)
Q-1 Find the eigen value and eigen vactor for the transpose of a matrix A where
A = 5 3
1 3
Q-2 Find the eigen value and eigen vactor for the Inverse of a matrix A where
A = 5 3
1 3
Q-3 Find the eigen value and eigenvectors of the matrix
4 2 2
5 3 2
2 4 1
A
52
Q-4 Find the eigen value and eigenvectors of the matrix
1 6 4
0 4 2
0 6 3
A
53
Q-5 Find the eigen value and eigenvectors of the matrix
0 1 0
0 0 1
1 3 3
A
Q-6 Find the eigen value and eigenvectors of the matrix
0 1 1
1 0 1
1 1 0
A
54
Q-7 Find the eigen value and eigenvectors of the matrix
0 0 2
1 2 1
1 0 3
A
Q-8 Find the eigen value and eigenvectors of the matrix
3 10 5
2 3 4
3 5 7
A
55
56
Tutorial – 10 eigen value&eigen vector (UNIT 10)
Q-1 Find eigen value and algebraic and geometric multiplicity of the following matrices.
(a)
0 0 1
1 0 3
0 1 3
(b)
1 2 2
0 2 1
1 2 2
(c)
1 0 0
0 1 0
0 0 1
Q-2 Verify Cayley-Hamilton thm for the following matrices.
57
(1) 3 1
4 2
(2) 5 4
1 2
(3)
2 1 1
1 2 1
1 1 2
(4)
1 1 2
1 2 1
0 1 1
Q-3 Using Cayley-Hamilton thm , Find 1 2 3 81 2
, , ,1 1
A A A A forA
58
Q-4 Given A=
2 3 1
3 1 2
1 2 3
. Using Cayley-Hamilton Thm compute
7 6 4 3 23 3 2 3A A A A A I
59
Q-5 Find the characteristic equation of the matrix
2 1 1
0 1 0
1 1 2
A
and hence find the
matrix represented by 8 7 6 5 4 3 25 7 3 5 8 2A A A A A A A A I
60
61
Tutorial – 11 MATHS 2 eigen value&eigen vector (UNIT 10) Q-1 Defn of complex matrix, Conjugate Matrix, Hermition, Skew-Hermition and Unitary
matrix and thm for Eigen value of these types of matrices.
Q-2 Show that the matrix 3 1 2
1 2 1
iA
i
is Hermition and Find nature of their Eigen
values.
62
Q-3 Show that the matrix 3
3 2
i iB
i i
is Skew Hermition and Find nature of their
Eigen values.
Q-4 Show that the matrix
3
2 2
3
2 2
i
Ci
is Unitary and Find nature of their Eigen
values.
63
Q-5 Show that
1 2 3
3 2 2 1
2 3 1 0
i i i
A i i
i
is a skew-Hermition and iA is a Hermition
matrix.
Q-6 Show that 1 11
1 12
i i
i i
is a Unitary matrix.
64
Q-7 Express the matrix
3 4 3
2 2
2 1 2
2 2
i i
i i
as a sum of Hermition and Skew-Hermition
matrices.
65
Q-8 Find the eigenvalues and bases for the eigenspaces of 25A and 2A I where
3 0
8 1A
Q-9 Show that cos sin
sin cosA
has no real eigenvalues and hence no eigenvectors
66
Q-10 Let be an eigenvalue of a matrix A. Then prove that (a) +k is an eigenvalue of
A+KI (b) K is an eigen value of KA.
67
Tutorial – 12 Inner product space (UNIT 9)
Q-1 Defn of Orthogonal and orthonormal set.
Q-2 Let 1V =(3,0,-1), 2V =(0,-1,0), 3V =(3,0,9) In 3R . Then
(a) Show that They form an orthogonal set under the standard Euclidean inner product
but not an orthonormal set
(b) Turn them into orthonormal set
Q-3 Let V= 2P . 2 2
0 1 2 0 1 2 0 0 1 1 2 2( ) & ( ) & ,p x a a x a x q x b b x b x p q a b a b a b Then
Generate orthonormal basisof V.
68
Q-4 1 2 3
4 3 3 4(0,1,0) & ( ,0, ) & ( ,0, )
5 5 5 5V V V
be an orthonormal basis for 3R with the
Euclidean inner product. Express the vector u=(1,1,1) as a linear combination of the
vectors 1V , 2V , 3V .lso find the coordinate vector w.r.t S={ 1V , 2V , 3V }
Q-5 Projection Theorem
69
Q-6 Let 3R have the Euclidean inner product and Let W be the subspace spanned by
orthonormal vectors 1 2
4 3(0,1,0) & ( ,0, )
5 5V V
, Find the orthogonal projection of
u=(1,1,1) on W. Also find the component of u orthogonal to W.
Q-7 Compute the orthogonal projection of u on a and the vector component of u
orthogonal to a for u=(-3,1) and a=(7,2)
Q-8 Gram Schmidt Process
70
Q-9 Using Gram Schmidt process orthonormalize the set of linearly independent vectors
1 2 3(1,0,1,1) & ( 1,0, 1,1) & (0, 1,1,1)u u u with standard inner product
Q-10 Using Gram Schmidt process orthonormalize the set of linearly independent vectors
1 2 3(1,1,1) & (0,1,1) & (0,0,1)u u u with standard inner product
71
Q-11 Consider the real inner product space 2P , where the polynomial
2 2
0 1 2 0 1 2 0 0 1 1 2 2( ) & ( ) & ,p x a a x a x q x b b x b x p q a b a b a b
For the basis 2 2 2
1 2 33 4 5 & 9 12 5 & 1 7 25u x x u x x u x x .
72
Tutorial – 13 Based on parametrization of plane curves (UNIT 2)
(1) X= a cost, y=b sint where 0≤t≤π into Cartesian equation. Also sketch the
curve C.
(2) Convert the parametric equations into Cartesian equatin x=at , y=2at(a>0) for
all t. Also sketch the curve.
(3) Find the length of the arc of the curve y=logsecx from x=0 to 3
x
.
73
(4) Find the perimeter of astroid 2 2 2
3 3 3x y a .
(5) Find the length of the curve 3 3cos , sinx a y a in the first quadrant.
(6) Find the perimeter of the cardioid (1 cos )r a .
(7) Find the area of the surface generated by revolving the curve 3y x from x=0
to x=3 about the x-axis.
74
(8) Find the surface area of the sphere of radius a.
(9) Find the surface of the solid generated by the revolution of the curve x=a
cos3t, y=a sin3t about the x-axis.
75
ANS
(1) Ellipse 2 2
2 21
x y
a b (2) Parabola 2 4y ax (3) log(2 3) (4) ba (5) 3a/2
(6) 8a (7) 3
2(730) 127
(8) 24 a (9) 212 / 5a
76
Tutorial – 14 Vector differencial calculus(UNIT 1,4)
(1) Find the angle between the tangents to the curve 2 3, 2 ,x t y t z t at the
points 1t .
(2) A particle moves along the curve 3 21, , 2 5x t y t Z t where t is time. Find
the components of its velocity and acceleratin at time t=1 in the direction
2 3 6i j k .
(3) Find the unit normal vector to the surface 2 2 22 7x y z at (1,-1,2).
(4) Find the angle between the surfaces 2 2 2 2 29 3x y z and x y z at the
point (2,-1,2).
77
(5) Find the directional derivative of the function 2 3( , , )f x y z xy yz at the point
(2,-1,1) in the direction of the vector i+2j+2k.
(6) If r xi yj zk
and | |r r
the snow that,
(a) r r
(b) 2( )n nr nr r
(c)
^
(log )r
rr
(7) Find divergence of the vector 2 2 33 5v x i xy xyz k at the point (1,2,3)
78
(8) Find the value of so that the the vector ( 3 ) ( 2 ) ( 3 )v x y i y z j x k is
solenoidal
(9) If r xi yj zk
then show that
(a) 3div r
(b) 0curl r
(10) Prove that 2 2( 3 2 ) (3 2 ) (3 2 2 )y z yz x i xz xy j xy xz z k is
irrotational.
79
ANS
(1) 1cos (9 / ) (2) 18/7 (3) 12 9 16i j k (4) 1cos (8 / 3 21) (6) -11/3
(8) 80 (9) =-2
80
Tutorial – 15 Vector integral calculus(UNIT 3)
(1) If 2 2 3( ) ( 2 ) (3 3 )f t ti t t j t t k then find 1
0
( )f t dt .
(2) If _
2 ( 1)r ti t j t k and _
22 6s t i t k then Evaluate
(a) 2 _ _
0
.r s dt (b) 2 _ _
0
r s dt
(3) Evaluate: _ _
.c
F d r if _
2 2( ) 2F x y i xyj and C is the rectangle bounded by
y=0,x=a,y=b and x=0.
81
(4) Find the circulatin of _
F around the curve,where _
F yi zj xk and C is the
circle 2 2 1, 0x y z .
(5) Find the work done in moving a particle from A(1,0,1) to B(2,1,2) along the
straight line AB in the force field _
2 ( ) ( )F x i x y j y z k .
(6) Evaluate : ^ ^
.s
F nds where _
18 12 3F zi j yk and S is the surface of the
plane 2x+3y+6z=12 in the first octane .
82
(7) Evaluate: ^ ^
.s
F nds where _
24F xzi y j yzk and S is the surface bounded by
the planes x=0,x=1 y=0,y=1,z=0,z=1.
(8) If _
2(2 3 ) 2 4F x z i xyj xk then evaluate _
V
F dV where V is bounded by
he planes x=0,y=0,z=0 and 2x+2y+z=4.
83
(9) Verify Green’s theorem in the plane for 2 2(3 3 ) (4 6 )C
x y dx y xy dy where C
is the boundry of the region bounded by 2 2&y x y x .
(10) Verify stoke’s theorem for _
2 2( ) 2F x y i xyj taken around the
rectangle bounded by the lines x= a, y=0, y=b.
(11) Verify divergence theorem for _
2 2 2( ) ( ) ( )F x yz i y xz j z xy k
taken over the rectangular parallelopiped 0≤x≤a, 0≤y≤b, 0≤z≤c.
84
85
ANS
(1) 1 2 7
2 3 4i j k (2) (a) 12 (b)
40 6424
3 5i j k (3) 22ab (4) - (5) 16/3
(6) 24 (7) 3/2 (8) 8/3 (9) 3/2 (10) -4a 2b (11) abc (a+b+c)
86
Tutorial – 16 Linear transformation(UNIT 8) (1) Check whether the following transformation are linear or not .
(1) T(x,y) = (x+y , xy) ,where T : R2 →R2 ,
(2) T(x,y) = (x+1 , y) , where T : R2→R2,
(3) T(x,y) = (x2 , x+y) ,where T : R2→R2,
(4) T(x,y) = (y , 3x-2y+1) ,where T : R2→R3 .
(2) Example of finding general formula.
(1) Let T : R3→R2 be a linear transformation . Assume T(1,1,1) = (1,2) ,
T(1,1,0) = (3,4) ,T(1,0,0) =(5,6),then find T(x,y,z).
87
(2) Let V1 =(1,1,1) , V2 = (1,1,0) , V3 = (1,0,0) . Let T : R3→R3 be a linear
transformations ,such that T(V1) = (2,-1,4) , T(V2) = (3,0,1), T(V3) = (-1,5,1)
.Find the formula for T(x,y,z) and use it to find T(2,4,-1).
88
( 3) Do as directed .
(1) Let T : R2→R2 be defined by T(x,y) = (x+y,x-y), prove that T is bijective .
(2) For each of the following linear transformation T, find range (T) and Ker(T)
(i) T:R2→R2 defined by T(x,y) = (x+y , x-y).
(ii) T:R3→R3 defined by T(x,y,z) = (2x-y+z , -x+3y+5z , 10x-9y+7z).
(3) Let T1 : M22→R and T2 : M22→M22 be the linear transformations given by
T1(A) = tr(A)
And T2(A) = AT then,
(i) Find (T10T2)(A) ,where A = 𝑎 𝑏𝑐 𝑑
(2) Can you find (T20T1)(A)?
89
90
(4) Let T1 : R
2→R2 and T2 : R2→R2 be the linear operators given by the
formula T1(x,y)=(x+y,x-y) and T2(x,y)=(2x+y,x-2y) ,
Show that (1)show that T1 and T2 are one one and (ii)verify that (T20T1)-
1=T2-10T1
-1.
91
ANSWERS :
EX. 1 : (1) NO (2) NO (3) NO (4) NO EX.2 : (1) T(X,Y,Z) = (5X-2Y-2Z , 6X-2Y-2Z)
(2)T(X,Y,Z) = (-X+4Y-Z,5X-5Y-Z,X+3Y) and T(2,-4,1) = (15,-9,-1)
EX.3 : (2) (i) range T = R2 , kernel T = {(0,0)} (3) a+d ; (T20T1)(A)
does not exists
(ii)range T = R3 , kernel T = {(0,0,0)}