E S S E N T IA L

s t e v e n b a r r y

+S t e p h e n

d a v i s

UNSWPRESS

MATHEMATICALSKILLS

for engineering, science and applied

mathematics

A UNSW Press book

Published by University of New South Wales Press Ltd

University of New South Wales UNSW Sydney NSW 2052

A U S TR ALIA www.unswpress.com.au

Steven Ian Barry and Stephen Alan Davis 2002 First published 2002

This book is copyright. Apart from any fair dealing for purposes of private study, research, criticism or

review, as permitted under the Copyright Act, no part may be reproduced by any process without written

permission. Inquiries should be addressed to the publisher.

National Library of Australia Cataloguing-in-Publication entry:

Barry, Steven Ian.Essential mathematical skills for engineering, science

and applied mathematics.

Includes index.ISBN 0 86840 565 5.

1. Mathematics. 2. Mathematics Problems, exercises, etc. I. Davis, Stephen. II. Title.

510

Printer BPA

vi

3 Transcendental Functions 313.1 Exponential Function...................................................................................................................313.2 Index L a w s .............................................................................................................................. .....323.3 Logarithm Rules .................................................................................................................... .....333.4 Trigonometric Fu nctio ns....................................................................................................... .....353.5 Trigonometric Identities ....................................................................................................... .....363.6 Hyperbolic Functions...................................................................................................................383.7 Example Q uestions................................................................................................................. .....39

4 Differentiation 414.1 First Principles .............................................................................................................................414.2 Linearity..........................................................................................................................................424.3 Simple D erivatives................................................................................................................. .....434.4 Product Rule ........................................................................................................................... .....434.5 Quotient R u le ........................................................................................................................... .....444.6 Chain Rule .............................................................................................................................. .....454.7 Implicit Differentiation.......................................................................................................... .....464.8 Parametric Differentiation .........................................................................................................474.9 Second Derivative.................................................................................................................... .....474.10 Stationary P o in ts .................................................................................................................... .....484.11 Example Q uestions................................................................................................................. .....50

5 Integration 515.1 Antidifferentiation ................................................................................................................. .....515.2 Simple Integrals.............................................................................................................................525.3 The Definite In te g ra l...................................................................................................................535.4 Areas ........................................................................................................................................ .....555.5 Integration by Substitution.........................................................................................................565.6 Integration by P a r ts ................................................................................................................. .....575.7 Example Q uestions................................................................................................................. .....58

6 Matrices 596.1 Addition..................................................................................................................................... .....596.2 Multiplication........................................................................................................................... .....606.3 Id e n tity ..................................................................................................................................... .....616.4 Transpose.......................................................................................................................................626.5 Determinants........................................................................................................................... .....63

6.5.1 Cofactor Expansion.........................................................................................................646.6 Inverse........................................................................................................................................ .....65

6.6.1 Two by Two Matrices................................................................................................ .....666.6.2 Partitioned Matrix .........................................................................................................666.6.3 Cofactors M a tr ix ....................................................................................................... .....67

6.7 Matrix M anipulation...................................................................................................................686.8 Systems of Equations...................................................................................................................706.9 Eigenvalues and Eigenvectors................................................................................................ .....736.10 Trace ........................................................................................................................................ .....746.11 Symmetric Matrices................................................................................................................. .....74

CONTENTS Vii

6.12 Diagonal Matrices.................................................................................................................... 746.13 Example Q uestions................................................................................................................. 75

7 Vectors 777.1 Vector N otation........................................................................................................................ 777.2 Addition and Scalar Multiplication...................................................................................... 787.3 Length........................................................................................................................................ 797.4 Cartesian Unit Vectors .......................................................................................................... 807.5 Dot Product.............................................................................................................................. 807.6 Cross Product........................................................................................................................... 827.7 Linear Independence.............................................................................................................. 837.8 Example Questions................................................................................................................. 86

8 Asymptotics and Approximations 878.1 L im its ........................................................................................................................................ 878.2 LHopitals R u le........................................................................................................................ 888.3 Taylor Series ........................................................................................................................... 888.4 Asymptotics.............................................................................................................................. 898.5 Example Questions................................................................................................................. 90

9 Complex Numbers 919.1 Definition.................................................................................................................................. 919.2 Addition and Multiplication ................................................................................................ 929.3 Complex Conjugate................................................................................................................. 929.4 Eulers Equation........................................................................................................................ 939.5 De Moivres Theorem.............................................................................................................. 949.6 Example Questions................................................................................................................. 95

10 Differential Equations 9710.1 First Order Differential Equations ...................................................................................... 97

10.1.1 Integrable.................................................................................................................... 9710.1.2 Separable.................................................................................................................... 9810.1.3 Integrating Factor....................................................................................................... 99

10.2 Second Order Differential Equations................................................................................... 10010.2.1 Homogeneous .......................................................................................................... 10010.2.2 Inhomogeneous.......................................................................................................... 102

10.3 Example Questions................................................................................................................. 105

11 Multivariable Calculus 10711.1 Partial Differentiation.............................................................................................................. 10711.2 Grad, Div and C u r l ................................................................................................................. 10811.3 Double Integrals........................................................................................................................ I l l11.4 Example Questions................................................................................................................. 114

CHAPTER 6MATRICES

6.1 ADDITION

If A and B a i ' e m x t i matrices such that

a n a i2 0>l n &n b\2 bln0*21 C122 &2 n &21 &22 t>2n

A = and B

ml m2 &mn bjni bm2 bmn

o-11 + &11 0-12 + bi2 din t 1^ n&21 + &21 0,22 + &22

60 MATRICES

6.2 MULTIPLICATION

A B is defined if A is size m x r and B size r x n. If

a n O-12 ^lm ' bn bl2 6i r

A -21 a 22 2^ 771

and B &21 &22 &2r

ari Ctr2 arTn ^nl &n2 bnr

then A B C is an rn x n matrix, where C\j anbij + a,i2b2j + ----- 1- a,irbrj. That is,Cij is the dot product of row i of A and column j of B .

In general A B / B A , that is, matrices are nonconunutative.

EXAMPLES

1 .1 23 4

3 3 7 5

' 1 - 1 ' ' 1 2 ' 1 to 1 to

1 2 3 4 7 K)

' 4 2 9 ' ' - 2 2 - 4 '3. If A - 3 - 1 1 and B 0 0 1

2 1 2 1 OO 1 1 1

l____

____

____

A B -2 9

- 1 1 1 2

- 8 + 0 + 27 6 + 0 + 3 -

- 4 + 0 + 6

19 - 2 8 - 2 3 9 - 1 0 10 2 - 4 - 9

- 2 0 3

8 + 0 - -6 + 0

4 + 0

20

- 4

36- 4

-4 1

- 1

-16 + 2 - 9 1 2 - 1 - 1

- 8 + 1 - 2

IDENTITY 61

while

B A -- 2 2 - 4

0 0 1 3 - 4 - 1

- 8 - 6 - 8 - 0 + 0 + 2

12 + 1 2 - 2

-2 2 2

22

4 2 9 - 3 - 1 1

2 1 2

-1019

- 4 - 2 - 4 0 + 0 + 1 6 + 4 - 1

-24 2

21

-18 + 2 - 8 0 + 0 + 2

2 7 - 4 - 2

4.

' 1 2 ' 1 ' 7 '1 1 1Q - 41 0 O 1

6.3 IDENTITY

The identity matrix, defined only for square matrices (n x n ), is

' 1 0 0 '0 1 0

I -

0 0 1

and is defined such that, for all n x n matrices A,

IA - >

11 II >

EXAMPLE

The 2 x 2 and 3 x 3 identity matrices are

1 0 00 1 00 0 1

62 MATRICES

6.4 TRANSPOSE

The transpose of a matrix is formed by writing its columns as rows. The transpose of an m x n matrix A is an n x m matrix denoted by A*, that is, if

a n ai2 a in a n a2i a-mi

A -21 a22 a2n

then A* -ai2 a22 dm 2

a-mi am2 amn a in a2n amn

EXAMPLES

1. If A then A* 0 2 1 1 4 - 1

' 4 2 9 ' ' 4 - 3 2 '2. If A - 3 - 1 1 then A* 2 - 1 1

2 1 2 9 1 2

If A and B are matrices and c is a scalar, then

1. (A*)* A

2. (A + B)* A* + B*

3. (cA)* - cA*

4. (A B )f = B* A*

EXAMPLE

(A1 A)1 = (A*(At)t) = AtA

DETERMINANTS 63

6.5 DETERMINANTS

The determinant of a 2 x 2 matrix A a b ' isc d

det(A) |A| a d - b e .

a n a i 2 O-1 3The determinant of a 3 x 3 matrix A 0 -2 1 0 -2 2 0 -2 3 is

0 -3 1 0 -3 2 0 -3 3

0 -2 2 0 -2 3 0 -2 1 0 -2 3+ & 1 3

2 1 0 -2 2 a i 2

0 -3 2 0 -3 3 0 -3 1 0 -3 3 0 -3 1 0 -3 2

(expanding by the first row).

EXAMPLES

1 . 1 23 4 4 6 2 ,

1 2 2 - 1 = - 1 - 4 = - 5 .

2 15 6

-3 1

_ Q 5 6 o 4 6 + 14 5 O - 3 1 z 2 1 2 - 3

- -3 ( 5 + 18) - 2(4 - 12) + ( -1 2 - 10)- - 7 5

3.1 7 5 o c0 2 6 - 1

z

0Do

0 0 OO

o

1 x 2 x 3 6

1 7 5 0 2 6 is not possible.

64 MATRICES

6 .5 .1 COFACTOR EXPANSION

The determinant of an n x n matrix may be found by choosing a row (or column) and summing the products of the entries of the chosen row (or column) and their cofactors:

det(A) a i jC ij + + + anjC nj,

(cofactor expansion along the j th column)

det(A) OtjiCji + Cli2Ci2 + +

INVERSE 65

in turn remembering to multiply by (

C n + 1

Cl2 = ( - 1 )

C13 + 1

0 2 1 0

0 2 1 0

0 0 1 1

- - 2

- 2

0

and so one, giving

C -- 2 2 0

3 - 3 14 - 2 0

6.6 INVERSE

A square matrix A is said to be invertible if there exists B such that

A B B A I.

B is denoted A -1 and is unique.

If det(A) 0 then a matrix is not invertible.

EXAMPLE

The matrix B

and

' 3 5 ' is the inverse of A 2 - 5 '1 2 - 1 3

A B -

B A -

' 3 5 ' 2 - 5 ' ' 1 0 '1 2 1 OO 0 1

2 --5 ' OO 5 ' ' 1 0 '

l 1 1 CO 1 2 i o 1

smce

- I

- I .

6 6 MATRICES

6 .6 .1 TWO BY TWO MATRICES

For 2 x 2 matrices, if A a b c d then

A - 1 -1 d b providing ad be ^ 0.ad be c a

If det(A) = ad bc 0 then A 1 does not exist

EXAMPLES

1. If A

2. If A

1 23 4

1 2 0 3

then A 1 - 2

then A -

4 - 2 - 3 1

- 2 '1

6 .6 .2 PARTITIONED MATRIX

Inverses can also be found by considering the partitioned matrix

A : 1

then performing row operations until the final partitioned matrix is of the form

I : A 1

EXAMPLE

The inverse of

1 2 10 1 01 1 0

can be calculated using row reductions where R3 - R3 R1 means that Row 3 becomes the old

INVERSE 67

Row 3 minus Row 1.

' 1 2 1 1 0 0 '0 1 0 0 1 01 1 0 0 0 1

i23 y R3 R 1

' 1 2 1 1 0 0 '0 1 0 0 1 0

1 o - 1 - 1 - 1 0 1

R3 y R3 -I- R2

1 2 1 1 0 00 1 0 0 1 00 0 - 1 - 1 1 1

R l -> R l - 2R2 R3 y R3

1 0 1 1 1 to 0

0 1 0 0 1 0

0 0 1 1 - 1 - 1

R l -> R l - R3

' 1 0 0 0 - 1 1 '0 1 0 0 1 0

1 o 0 1 1 - 1 - 1

hence

' 1 2 1 ' -1 ' 0 - 1 1 '0 1 0 - 0 1 01 1 1

o

1 - 1 - 1

6 .6 .3 COFACTORS MATRIX

The inverse of a n x n matrix A can be found by considering the transpose of the cofactors matrix divided by the determinant:

where C\j is the determinant of A with row i and column j deleted, multiplied by (1)*+,\ The matrix C is called the cofactors matrix.

6 8 MATRICES

EXAMPLES

1. If

then

A

C u

1 2 10 1 01 1 0

1 01 0

- 1 )21

0

10

- 1

and so on. Since | A| 1 we get

A - = r j

0 0 - 1 'T

' 0 - 1 1 '

1 - 1 1 - 0 1 0

- 1 0 - 1 1 - 1 - 1

2. The matrix

A -

has cofactors matrix

C -

1 2 3 0 0 2 1 1 0

-2 2 0 3 - 3 1 4 - 2 0

hence the inverse

A -1 --2 3 2 - 3 0 1

6.7 MATRIX MANIPULATION

Matrices do not behave as real numbers. When manipulating matrix expressions a distinction is made between multiplying from the left (pre-multiplication) and multiplying from the right (post-multiplication).

MATRIX MANIPULATION 69

EXAMPLES

1. Given that A B C I find B ?

A B C (A 1 A) B C / B (C C -1 )

B

IA -1 IA - 1 I C - 1A - 1 C - 1(C A )-1.

pre-multiply both sides by A - 1 post-multiply both sides by C - 1 simplifying

2. If A PD P 1, then A 3 is

A 3 - (PDP- 1 ) (PDP- 1 ) (P D P - 1 ) P D (P - 1 P ) D P - 1 P D P - 1 P D 2P - 1 P D P - 1 P D 3P - 1

since P P - 1 = I again since P P - 1 I.

3. If A v Xv then

A 3v A A A v AA(Av) XAAv

X2A v

= X3v.

4. If A v Xv then if A 1 exists then

A - 1 A v A - 1 Xv AA- 1 v

=> v AA-1 v

A 1- v = A v.A ~ ^

See Section 6.9 on eigenvalues since this example shows that if A has eigenvalue A, with eigenvector v, then A - 1 has eigenvalue 1/A for the same eigenvector.

7 0 MATRICES

6.8 SYSTEMS OF EQUATIONS

Systems of m linear equations involving n unknowns may be written as a matrix equation. For example,

x + y + 2z 1 2x + Ay 3z 5 3x + 6y 5z 2

is written as

' 1 1 2 ' X ' 1 '2 4 - 3 y 53 6 - 5 z 2

A x b.

Systems of equations are typically solved by Gaussian elimination.

If A is invertible then x A _1b.

Gaussian Elimination allows

a multiple of one row to be added to another row.

a row to be multiplied by a (non-zero) number.

Hence R3 R3 2R1 means each element in Row 3 becomes the old Row 3 element minus two times the corresponding Row 2 element.

EXAMPLES

1. The augmented matrix is an easy way of writing systems of equations. For the following system

x + y + 2z 1 2x + Ay 3z 5 3x + 6y 5z 2

SYSTEMS OF EQUATIONS 71

the augmented matrix is

' 1 1 2 1 '0 2 - 7 30 3 - 1 1 - 1

' 1 1 2 1 '0 1 --7/2 3/2

_ 0 3 - 1 1 - 1

' 10

11 -

2-7/2 3/

_ 0 0 -- 1/2 - 1 1 /

' 1 1 2 1 '0 1 --7/2 3/20 0 1 1 1

R2 R 2 ~ 2i?i R$ R$ 3Ri

R$ ^ R$ 3R2

This gives the straightforward solution by back substitution of x 61, y 40, z 11.

2. Consider the system

2x 5y 2 x + 3y 4

written as A x b such that

A - 2 - 5 ' X ' - 2 '- 1 3 > x = . y ., - 4

The matrix A has inverse A 1 3 5 1 2 so

72 MATRICES

After performing Gaussian reduction by row operations the three cases (no solution, infinite solutions, one solution) are typically represented by the following:

1. If you perform row operations to obtain

a b c h '0 d e hi

_ 0 0 f k3

(where a , .., f are non-zero real numbers) then you get one unique solution.

2. If you perform row operations to obtain

a b c * i '0 d e k2

_ 0 0 0 h _

then if / 0 you get no solution.

3. If you perform row operations to obtain

a b c * i '0 d e hi

_ 0 0 0 0

then you get an infinite number of solutions that represent a line where you let z t, t is some parameter, and then express x, y in terms of t.

EXAMPLE

To solve the system:

' 1 - 2 - 1 ' X ' - 1 '2 1 3 y - 131 8 9 z 29

perform row reductions to obtain

' 1 1 to - 1 - 1 '0 1 1 00

1 o 0 0 1o

and setting z t gives y 3 t and x 2(3 t) t 1 so

(x, y, z) = (5 - t, 3 - t, t) = (5,3,0) + t ( - 1, - 1 , 1 )

or a line in three dimensional space.

EIGENVALUES AND EIGENVECTORS 73

6.9 EIGENVALUES AND EIGENVECTORS

If A is an n x n matrix then a scalar A is called an eigenvalue of A, if associated with it there is a non-zero vector v, called an eigenvector, such that

A v Xv.

To find the eigenvalues solve the characteristic equation

| A - AI| - 0.

To find the eigenvectors solve

(A - XI)v = 0.

EXAMPLE

To find the eigenvalues and eigenvectors for

A -

set up the characteristic equation

det

0 1 1 0

' 0 1 ' ' 1 0 ' \ -A 11 0

A0 1 ) = 1 -A = 0 ,

which gives A2 1 0 so Ai 1 and A2 = 1 are the eigenvalues. To find the eigenvectors solve

v 0 .-A 11 -A

For Ai 1 let

v i - " Xi then' - 1 1 ' ' Xi _ ' o '

. y i . 1 - 1 . y i . 0

Both equations give xi yi 0 so y\ is a free variable. Hence the eigenvector corresponding to Ai 1 is t( 1,1), where t is any number, t e Re.For A2 1 let

74 MATRICES

Both equations give 2 + 2/2 0 so 2/2 is a free variable. Hence the eigenvector corresponding to A2 1 is p( 1, 1). The length of the eigenvector is unimportant hence it is convenient to write

i = ( l , l ) , 2 = (1 , - 1 ).

6.10 TRACE

The trace of a matrix is the sum of its diagonal elements.(Note that the trace is also equal to the sum of the eigenvalues.)

EXAMPLE The trace of1 2 3 4 5 6 7 8 9

1 + 5 + 9 15.

6.11 SYMMETRIC MATRICES

The matrix A is symmetric if A A 4.

EXAMPLE The matrix1 2 32 5 63 6 9

is symmetric.

6.12 DIAGONAL MATRICES

A diagonal matrix is one with only terms along the main diagonal.

EXAMPLE A 3 x 3 diagonal matrix has the forma 0 0 0 6 0 0 0 c

EXAMPLE Q UESTION S 75

6.13 EXAMPLE QUESTIONS(Answers are given in Chapter 14)

1. Find A + B , A B , B A and the trace(A):

(i)

A =4 02 31 0

1- 7

01 - 1 - 1

- 2 - 2 - 23 1 7

(ii)

(iii)

A =

B =

A =

2. FindA B:

(i) A =

(ii) A =

B =

1 96 4

6 8- 7 1

2 0 00 3 00 0 4

9 0 00 8 00 0 - 3

93 B =2

- 1 1 _

6- 2(iii) A =

3. Find A*, A* A, A A 4:

(i) A=

2 1 16

b = r i - i

2 - 1 34 3 - 5

(ii) A=

4. Find the inverse, if it exists, of the matrices

(i)

(ii)

5. Find the determinants of the following matrices.

1 0 1 03 2 - 7 1

(i) 6 1 - 6 12 2 2 3

' 4 1 6 20 1 4 2

(ii) 0 0 9 00 0 0 - 1

' 3 0 4 '(iii) 1 0 0

0 0 6

Solve the system of equations

X + 2 y z = 2

8x + 3y - 7 z = 4i y - 12z = - 8

First showing that a non-trivialsolve

4x --y =4 y - z =

4x + 17y 4z =

8. For what values of a and c do you get

(i) one solution,

(ii) no solution,

(iii) infinite solutions,

for the system

x + 5y + z = 0 x + 6y z = 2

2x + ay + z = c ?

9. Find the eigenvalues and eigenvectors of A:

(i)

6 0 ' 2 0 10 - 4 A = 0 3 47 5 0 0 1

(ii)2 3 2 1 01 4 A = 1 2 01 2 0 ' 0 0 10 1 00 3 2

76

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CHAPTER 7VECTORS

7.1 VECTOR NOTATION

EXAMPLES

1. The vector (1,0) points in the x direction and has length 1.

2. The vector (1,1) points in a direction with angle 7r/4 to the x axis.

78 VECTORS

A vector in R n is represented by an ordered n-tuple

v = (vi,v2, . . . , v n).

EXAMPLES

1. v (1,3,4) is a three dimensional vector so v e R 3.

2. v (3 ,5 ,7 ,1 , 2) is a five dimensional vector so v e R 5.

7.2 ADDITION AND SCALAR MULTIPLICATION

If v (vi,v2 , . . . ,vn), w (wi,w2, . .. ,wn) and c is a scalar constant then

V + W (t>i + Wi, V2 + W2, . . , vn + wn)

CV - (0Vi,CV2,...,CVn).

EXAMPLES

1. If v (1, 1,4) and w (1, 3,3) then,

S + = (2> - 4 >7)>

and4v - (4 , - 4 , 1 6 ) .

2. If i (1,0) and j = (0,1) thenv = (2,3) = 2 i + 3 j .

3. If v ( l ,x ,2 ,x ) and u ( 2 ,\,x, 0) then v + u (3,1 + x, 2 + x,x).

4. (1,2) + (4 ,5 ,6) is not defined.

LENGTH 79

7.3 LENGTH

The length of a vector in R n is given by

IMI = \ j v l + v l + - - - + v l .

EXAMPLES

1. ||(1,2,1)|| = V l 2 + 22 + l 2 = -s/6

2 . ||(1,1 ,2 ,1 ,3)|| = >/1 + 1 + 4 + 1 + 9 = >/16 = 4

The triangle inequality states that

||u + II < ||u|| + ||||.

That is the length of the sum of vectors must be less than the length of the two individual vectors added.

EXAMPLE

(0,3) + (4,0) - (4,3) and ||(0,3)|| - 3, ||(4,0)|| - 4,||(4,3)|| - V ^ + 4 * - 5 < 3 + 4.

y

80 VECTORS

7.4 CARTESIAN UNIT VECTORS

The Cartesian unit vectors for R 3 are

(1)0,0), j = (0,1,0), f e - (0,0,1).

Vectors in R 3 are often written as the sum of the components in the direction of the Cartesian unit vectors:

V - ( V I , V 2 , V 3 ) - v i i + v 2 j + v 3 k .

EXAMPLES

1. (1,2,3) = i + 2 j + 3k

2 . (0, 2 , 0) = 2j

7.5 DOT PRODUCT

If u and v are vectors in R n then the dot product is defined by

U V U i V i + U2 V2 H------------- 1- u n v n .

This is also called an inner product on R n. The result of a dot product is a scalar.

EXAMPLES

1. (1,2,3) (1,1,1) 1 + 2 + 3 6

2. (1,2,3) (1,2,3) l 2 + 22 + 32 14

3. IImII2 u u

DOT PRODUCT 81

The angle 6 between two vectors is given by

u vcos 6 ,,

u lit)

EXAMPLES

1. The angle 6 between (1,2,3) and (1,1,1) is such that

6 6COS u . -------- = .V l2 + 22 + 32 V l2 + l 2 + l 2 7 4 2

2 . (1 , 2 ,3) and (1 , 1 , 1 ) are at right angles since (1 , 2 ,3) (1 , 1 , 1 ) 0 hence cos# 0.

Two vectors, u and v are orthogonal if they are perpendicular to each other and

u v 0.

EXAMPLES

1. u (1 ,2 , 1 ) and v (2 , 1 , 4) are perpendicular since u - v 2 + 2 4 0.

2. To find a vector, (a, b), perpendicular to (1,2) write

(a, 6) (1 , 2) a + 26 0

hence the simplest choice is (a, b) (2 , 1 ) although any multiple of this will be perpendicular to (1,2). For example (2, 1) and (4,2) are still perpendicular to (1,2).

82 VECTORS

7.6 CROSS PRODUCT

If u and v are two vectors in R 3, then the cross product u x v is defined in determinant notation by

i j kU X V U i U2 U3

V i V2 V3

iU2 U3 ~ j

U i U 3 + k U i U2V2 V3 VI V3 V i V2

( U2 V3 U3 V2 , U 3 V1 U 1 V3 , U 1V2 U2 V1 ) .

EXAMPLES

1. (1 ,0 ,0) x (0,1,0) - (0,0,1) . That is i x j = k .

2. If u (2, 3,1) and v (12,4, 6), then

i 3 h

W U X V 2 - 3 112 4 - 6

- 3 1 2 1 2 - 3i 4 - 6 - 3 12 - 6 + fe 12 4

- (18 - 4) - j ( - 12 - 12) + fe(8 + 36)

= 14 i + 24 j + 44 fc = (14,24,44).

3. The cross product (1,2,3) x (1,0,1) is:

(1,2,3) x (1,0,1)

4. The cross product (1,1,2) x (1, 1,0) is:

(1 , 1 , 2) x (1 , - 1 , 0) =

j k

1 2 3 1 0 1

(2 , 2 , 2).

i j k

1 1 2 1 - 1 0

(2 , 2 , 2).

LINEAR INDEPENDENCE 83

Note that the result of taking the cross product of two vectors is another vector where the direction of u x v is perpendicular to both u and v.

EXAMPLES

1. In a previous example (2, 3,1) x (12,4, 6) (14,24,44) and

(2,3, - 1 ) (14,24,44) = 28 - 72 + 44 = 0.

Similarly (12,4, - 6) (14,24,44) - 0.

2. In a previous example (1,1,2) x (1, 1,0) (2,2, 2). Note that (2,2, 2) (1,1,2) 0 and (2 , 2 , 2) ( 1, 1, 0) 0 .

7.7 LINEAR INDEPENDENCE

A vector v is a linear combination of the vectors u, 2 j ., u n if it can be written as

V Cl t i l + C2W2 + + cnun

where c,... , c are constants.

EXAMPLES

1 . (2 ,7 ,3 ) is a linear combination of (1 , 1 , 0), (0, 2 , 1 ), (0, 1 , 0) since

( 2 , 7 , 3 ) - 2 ( 1 , 1 , 0 ) + 3 ( 0 , 2 , 1 ) - ( 0 , 1 , 0 ) .

2 . (1 , 2 , 1 ) is not a linear combination of ( 1 , 1 , 0), (2 , 1 , 0), (1 , 0, 0) since we can never combine the three vectors to get the third component of (1 , 2 , 1 ).

3. Any vector (a, b, c) in R 3 can be found from a linear combination of { (1 ,0 ,0 ) , (0,1,0) , (0 ,0 ,1)} .

84 VECTORS

A set of vectors u, 2 , , u n are linearly independent if the only constants c i ,. that satisfy

Cl t i l + C22 H-------h C n U n 0

are ci C2 c 0.

j cn

EXAMPLES

1. ( 1 , 1 ,0 ) , (0 ,2 ,1) , (0 ,1,0 ) are independent since

ci ( 1 , 1 ,0 ) + c2 (0 ,2 ,1) + c3(0 ,1,0 ) (0,0,0)

implies

ci 0

ci + c2 + c3 0

C2 0

which gives ci C2 C3 0.

2. ( 1 ,1 , 0), (2 ,1, 0), (1 ,0,0) are dependent (not linearly independent) since

ci + 2c2 + C3 0ci + c2 0

0 0

has an infinite number of solutions, one being ci 1 , C2 1 , C3 1.

3. The vectors (1 ,2), (2, 1), (1 ,0) are dependent since

c i( l , 2) + c2(2, 1) + c3(l, 0) (0,0)

implies

Ci + 2c2 + C3 02ci -(- 2c2 = 0.

Since we have two equations in three unknowns we can always find a non-zero a , c2, C3 to satisfy these equations, for example c 1, c2 1, C3 1. More than two vectors in R2 can never be independent.

4. The vectors i , j , k are independent since for any vector v (a, b,c) it is possible to write

(a, b, c) ci i + c2j + C3k ai + bj + ck

hence if v 0 then ci c2 C3 0.

LINEAR INDEPENDENCE 85

A set of vectors is linearly independent if the determinant of the matrix with vectors as columns is not zero.

EXAMPLES

1. For (1,1,0) , (0,2,1) , (0,1,0) the determinant

1 0 0 1 2 10 1 0

- - I / O

hence the vectors are independent.

2. For (1,1,0) , (2,1,0) , (1,0,0) the determinant

1 2 11 1 00 0 0

- 0

hence the vectors are dependent. We can show that

(2 , 1 , 0) - ( 1 , 1 , 0) + (1 , 0, 0)

so they are not independent of each other.

86 VECTORS

7.8 EXAMPLE QUESTIONS(Answers are given in Chapter 14)

1. Evaluate the sum u + v , 3 u and ||u||:IV IV IV IV

(i) u = ( - 2 , - 1 ) , = (1 ,1)

(ii) u = (3 ,4 ) , = (4,3)

(iii) u = (-2,1), v = (-1, -1)

(iv) u = ( 3 ,4 ,2 ) , = (1 ,1 ,1 )

(v) u = (3 ,1 ,1 ,0 ) , = (1 ,0 ,1 ,1 )

(vi) u = 2 i + 3 j + k , v = i j kv v iv iv iv iv(vii) u = i + j , v = i 3 j

V V ^ V V ^

2. For the above vectors verify the triangle inequality that||u + || 2 C 3 a ii> 3 C 2 aS2i>lC3

+ aS2i>3 Cl + 036 10 2 0 36 2 0 1.

12. Verify the above equation using the vectors a = ( 1 , 1 , 2 ) , b = ( 1 , 0 , 1 ) , c = (0 , 1 , 1) .