Schaum's Outlines Linear Algebra 6th Edition pdf pdf

Linear Algebra

Sixth Edition

Seymour Lipschutz, PhD

Temple University

Marc Lars Lipson, PhD

University of Virginia

Schaum’s Outline Series

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SEYMOUR LIPSCHUTZ _{is on the faculty of Temple University and formally taught at the Polytechnic}

In-stitute of Brooklyn. He received his PhD in 1960 at Courant InIn-stitute of Mathematical Sciences of New York

University. He is one of Schaum’s most prolific authors. In particular, he has written, among others, Beginning Linear Algebra, Probability, Discrete Mathematics, Set Theory, Finite Mathematics, and General Topology.

MARC LARS LIPSON _{is on the faculty of the University of Virginia and formerly taught at the University} of Georgia, he received his PhD in finance in 1994 from the University of Michigan. He is also the coauthor of Discrete Mathematics and Probability with Seymour Lipschutz.

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Preface

Linear algebra has in recent years become an essential part of the mathematical background required by mathematicians and mathematics teachers, engineers, computer scientists, physicists, economists, and statis-ticians, among others. This requirement reflects the importance and wide applications of the subject matter.

This book is designed for use as a textbook for a formal course in linear algebra or as a supplement to all current standard texts. It aims to present an introduction to linear algebra which will be found helpful to all readers regardless of their fields of specification. More material has been included than can be covered in most first courses. This has been done to make the book more flexible, to provide a useful book of reference, and to stimulate further interest in the subject.

Each chapter begins with clear statements of pertinent definitions, principles, and theorems together with illustrative and other descriptive material. This is followed by graded sets of solved and supplementary problems. The solved problems serve to illustrate and amplify the theory, and to provide the repetition of basic principles so vital to effective learning. Numerous proofs, especially those of all essential theorems, are included among the solved problems. The supplementary problems serve as a complete review of the material of each chapter.

The first three chapters treat vectors in Euclidean space, matrix algebra, and systems of linear equations. These chapters provide the motivation and basic computational tools for the abstract investigations of vector spaces and linear mappings which follow. After chapters on inner product spaces and orthogonality and on determinants, there is a detailed discussion of eigenvalues and eigenvectors giving conditions for representing a linear operator by a diagonal matrix. This naturally leads to the study of various canonical forms, specifically, the triangular, Jordan, and rational canonical forms. Later chapters cover linear functions and the dual spaceV*, and bilinear, quadratic, and Hermitian forms. The last chapter treats linear operators on inner product spaces. The main changes in the sixth edition are that some parts in Appendix D have been added to the main part of the text, that is, Chapter Four and Chapter Eight. There are also many additional solved and supplementary problems.

Finally, we wish to thank the staff of the McGraw-Hill Schaum’s Outline Series, especially Diane Grayson, for their unfailing cooperation.

SEYMOURLIPSCHUTZ

MARCLARSLIPSON

List of Symbols

AH, conjugate transpose, 38

AT, transpose, 33

Aþ, Moore–Penrose inverse, 420

Aij, minor, 271 AðI;JÞ, minor, 275

AðVÞ, linear operators, 176 adjA, adjoint (classical), 273

AB, row equivalence, 72

A’B, congruence, 362

C, complex numbers, 11

Cn_{, complexn-space, 13}

C½a;b, continuous functions, 230

CðfÞ, companion matrix, 306 colspðAÞ, column space, 120

dðu;vÞ, distance, 5, 243

HomðV;UÞ, homomorphisms, 176

i,j,k, 9

I_n, identity matrix, 33

Im F, image, 171

JðlÞ, Jordan block, 331

K, field of scalars, 112 KerF, kernel, 171

S?, orthogonal complement, 233 sgns, sign, parity, 269

spanðSÞ, linear span, 119 trðAÞ, trace, 33

½T_S, matrix representation, 197

T*, adjoint, 379

T-invariant, 329

Tt_{, transpose, 353}

kuk, norm, 5, 13, 229, 243

½u_S, coordinate vector, 130

uv, dot product, 4, 13

V**, second dual space, 352

Vr

V, exterior product, 403

W0_{, annihilator, 353} z, complex conjugate, 12

Zðv;TÞ,T-cyclic subspace, 332

d_ij, Kronecker delta, 37

DðtÞ, characteristic polynomial, 296

l, eigenvalue, 298

P

Contents

List of Symbols iv

CHAPTER 1 Vectors in Rnand Cn, Spatial Vectors 1

1.1 Introduction 1.2 Vectors in Rn 1.3 Vector Addition and Scalar Multi-plication 1.4 Dot (Inner) Product 1.5 Located Vectors, Hyperplanes, Lines, Curves inRn 1.6 Vectors inR3(Spatial Vectors),ijkNotation 1.7 Complex Numbers 1.8 Vectors inCn

CHAPTER 2 Algebra of Matrices 27

2.1 Introduction 2.2 Matrices 2.3 Matrix Addition and Scalar Multiplica-tion 2.4 SummaMultiplica-tion Symbol 2.5 Matrix MultiplicaMultiplica-tion 2.6 Transpose of a Matrix 2.7 Square Matrices 2.8 Powers of Matrices, Polynomials in Matrices 2.9 Invertible (Nonsingular) Matrices 2.10 Special Types of Square Matrices 2.11 Complex Matrices 2.12 Block Matrices

CHAPTER 3 Systems of Linear Equations 57

3.1 Introduction 3.2 Basic Definitions, Solutions 3.3 Equivalent Systems, Elementary Operations 3.4 Small Square Systems of Linear Equations 3.5 Systems in Triangular and Echelon Forms 3.6 Gaussian Elimination 3.7 Echelon Matrices, Row Canonical Form, Row Equivalence 3.8 Gaussian Elimination, Matrix Formulation 3.9 Matrix Equation of a System of Linear Equations 3.10 Systems of Linear Equations and Linear Combinations of Vectors 3.11 Homogeneous Systems of Linear Equations 3.12 Elementary Matrices 3.13LU Decomposition

CHAPTER 4 Vector Spaces 112

4.1 Introduction 4.2 Vector Spaces 4.3 Examples of Vector Spaces 4.4 Linear Combinations, Spanning Sets 4.5 Subspaces 4.6 Linear Spans, Row Space of a Matrix 4.7 Linear Dependence and Independence 4.8 Basis and Dimension 4.9 Application to Matrices, Rank of a Matrix 4.10 Sums and Direct Sums 4.11 Coordinates 4.12 Isomorphism of V and Kn 4.13 Full Rank Factorization 4.14 Generalized (Moore_–Penrose) Inverse 4.15 Least-Square Solution

CHAPTER 5 Linear Mappings 166

5.1 Introduction 5.2 Mappings, Functions 5.3 Linear Mappings (Linear Transformations) 5.4 Kernel and Image of a Linear Mapping 5.5 Singular and Nonsingular Linear Mappings, Isomorphisms 5.6 Operations with Linear Mappings 5.7 AlgebraA(V) of Linear Operators

CHAPTER 6 Linear Mappings and Matrices 197

6.1 Introduction 6.2 Matrix Representation of a Linear Operator 6.3 Change of Basis 6.4 Similarity 6.5 Matrices and General Linear Mappings

CHAPTER 7 Inner Product Spaces, Orthogonality 228

7.1 Introduction 7.2 Inner Product Spaces 7.3 Examples of Inner Product Spaces 7.4 Cauchy_–Schwarz Inequality, Applications 7.5 Orthogonal-ity 7.6 Orthogonal Sets and Bases 7.7 Gram–Schmidt Orthogonalization

Process 7.8 Orthogonal and Positive Definite Matrices 7.9 Complex Inner Product Spaces 7.10 Normed Vector Spaces (Optional)

CHAPTER 8 Determinants 266

8.1 Introduction 8.2 Determinants of Orders 1 and 2 8.3 Determinants of Order 3 8.4 Permutations 8.5 Determinants of Arbitrary Order 8.6 Proper-ties of Determinants 8.7 Minors and Cofactors 8.8 Evaluation of Determi-nants 8.9 Classical Adjoint 8.10 Applications to Linear Equations, Cramer’s Rule 8.11 Submatrices, Minors, Principal Minors 8.12 Block Matrices and Determinants 8.13 Determinants and Volume 8.14 Determinant of a Linear Operator 8.15 Multilinearity and Determinants

CHAPTER 9 Diagonalization: Eigenvalues and Eigenvectors 294

9.1 Introduction 9.2 Polynomials of Matrices 9.3 Characteristic Polynomial, Cayley–Hamilton Theorem 9.4 Diagonalization, Eigenvalues and Eigenvec-tors 9.5 Computing Eigenvalues and EigenvecEigenvec-tors, Diagonalizing Matrices 9.6 Diagonalizing Real Symmetric Matrices and Quadratic Forms 9.7 Minimal Polynomial 9.8 Characteristic and Minimal Polynomials of Block Matrices

CHAPTER 10 Canonical Forms 327

10.1 Introduction 10.2 Triangular Form 10.3 Invariance 10.4 Invariant Direct-Sum Decompositions 10.5 Primary Decomposition 10.6 Nilpotent Operators 10.7 Jordan Canonical Form 10.8 Cyclic Subspaces 10.9 Rational Canonical Form 10.10 Quotient Spaces

CHAPTER 11 Linear Functionals and the Dual Space 351

11.1 Introduction 11.2 Linear Functionals and the Dual Space 11.3 Dual Basis 11.4 Second Dual Space 11.5 Annihilators 11.6 Transpose of a Linear Mapping

CHAPTER 12 Bilinear, Quadratic, and Hermitian Forms 361

12.1 Introduction 12.2 Bilinear Forms 12.3 Bilinear Forms and Matrices 12.4 Alternating Bilinear Forms 12.5 Symmetric Bilinear Forms, Quadratic Forms 12.6 Real Symmetric Bilinear Forms, Law of Inertia 12.7 Hermitian Forms

CHAPTER 13 Linear Operators on Inner Product Spaces 379

13.1 Introduction 13.2 Adjoint Operators 13.3 Analogy BetweenA(V) and

C_{, Special Linear Operators 13.4 Self-Adjoint Operators 13.5 Orthogonal}

and Unitary Operators 13.6 Orthogonal and Unitary Matrices 13.7 Change of Orthonormal Basis 13.8 Positive Definite and Positive Operators 13.9 Diagonalization and Canonical Forms in Inner Product Spaces 13.10 Spec-tral Theorem

APPENDIX A Multilinear Products 398

APPENDIX B Algebraic Structures 405

APPENDIX C Polynomials over a Field 413

APPENDIX D Odds and Ends 417

Vectors in R

n

and C

n

,

Spatial Vectors

1.1

Introduction

There are two ways to motivate the notion of a vector: one is by means of lists of numbers and subscripts, and the other is by means of certain objects in physics. We discuss these two ways below.

Here we assume the reader is familiar with the elementary properties of the field of real numbers, denoted byR. On the other hand, we will review properties of the field of complex numbers, denoted by

C. In the context of vectors, the elements of our number fields are calledscalars.

Although we will restrict ourselves in this chapter to vectors whose elements come fromR and then fromC, many of our operations also apply to vectors whose entries come from some arbitrary field K.

Lists of Numbers

Suppose the weights (in pounds) of eight students are listed as follows:

156; 125; 145; 134; 178; 145; 162; 193

One can denote all the values in the list using only one symbol, sayw, but with different subscripts; that is,

w₁; w₂; w₃; w₄; w₅; w₆; w₇; w₈

Observe that each subscript denotes the position of the value in the list. For example,

w1 ¼156; the first number; w2¼125; the second number; . . .

Such a list of values,

w¼ ðw₁;w₂;w₃;. . .;w₈Þ

is called alinear array orvector.

Vectors in Physics

Many physical quantities, such as temperature and speed, possess only“magnitude.”These quantities can be represented by real numbers and are calledscalars. On the other hand, there are also quantities, such as force and velocity, that possess both _“magnitude_” and _“direction._” These quantities, which can be represented by arrows having appropriate lengths and directions and emanating from some given reference pointO, are called vectors.

Now we assume the reader is familiar with the spaceR3 where all the points in space are represented by ordered triples of real numbers. Suppose the origin of the axes inR3 is chosen as the reference pointOfor the vectors discussed above. Then every vector is uniquely determined by the coordinates of its endpoint, and vice versa.

There are two important operations, vector addition and scalar multiplication, associated with vectors in physics. The definition of these operations and the relationship between these operations and the endpoints of the vectors are as follows.

(i) Vector Addition: The resultantuþvof two vectorsuandvis obtained by theparallelogram law; that is, uþv is the diagonal of the parallelogram formed byuand v. Furthermore, if ða;b;cÞ and

ða0;b0;c0Þare the endpoints of the vectorsuandv, thenðaþa0; bþb0; cþc0Þis the endpoint of the vectoruþv. These properties are pictured in Fig. 1-1(a).

(ii) Scalar Multiplication: The productkuof a vectoruby a real numberkis obtained by multiplying the magnitude ofubyk and retaining the same direction ifk>0 or the opposite direction ifk<0. Also, ifða;b;cÞis the endpoint of the vectoru, thenðka;kb;kcÞis the endpoint of the vectorku. These properties are pictured in Fig. 1-1(b).

Mathematically, we identify the vectoruwith itsða;b;cÞand writeu¼ ða;b;cÞ. Moreover, we call the ordered tripleða;b;cÞof real numbers a point or vector depending upon its interpretation. We generalize this notion and call ann-tupleða₁;a₂;. . .;a_nÞof real numbers a vector. However, special notation may be used for the vectors inR3 calledspatial vectors (Section 1.6).

1.2

Vectors in R

n

The set of alln-tuples of real numbers, denoted byRn, is calledn-space. A particularn-tuple inRn, say

u¼ ða₁;a₂;. . .;a_nÞ

is called apoint or vector. The numbers a_i are called thecoordinates, components, entries, or elements

ofu. Moreover, when discussing the spaceRn, we use the termscalarfor the elements of R.

Two vectors,uandv, areequal, writtenu¼v, if they have the same number of components and if the

corresponding components are equal. Although the vectors ð1;2;3Þandð2;3;1Þcontain the same three numbers, these vectors are not equal because corresponding entries are not equal.

The vectorð0;0;. . .;0Þwhose entries are all 0 is called thezero vectorand is usually denoted by 0.

EXAMPLE 1.1

(a) The following are vectors:

ð2; 5_Þ; ð7;9_Þ; ð0;0;0_Þ; ð3;4;5_Þ

The first two vectors belong toR2, whereas the last two belong toR3. The third is the zero vector inR3. (b) Findx;y;zsuch thatðx y; xþy; z 1Þ ¼ ð4;2;3Þ.

By definition of equality of vectors, corresponding entries must be equal. Thus,

x y_¼4; x_þy_¼2; z 1_¼3

Solving the above system of equations yieldsx¼3,y¼ 1,z¼4.

Figure 1-1

z

y

x 0

u

ku

(a, b, c) (a, b, c)

(b) Scalar Multiplication

(ka, kb, kc) (a′, b′, c′)

z

y

x 0

v

u u +v

(a + _a′_{, b} + _b′_{, c} + _c′₎

Column Vectors

Sometimes a vector inn-spaceRn is written vertically rather than horizontally. Such a vector is called a

column vector, and, in this context, the horizontally written vectors in Example 1.1 are calledrow vectors. For example, the following are column vectors with 2;2;3, and 3 components, respectively:

1

We also note that any operation defined for row vectors is defined analogously for column vectors.

1.3

Vector Addition and Scalar Multiplication

Consider two vectorsuandv inRn, say

u_{¼ ð}a1;a2;. . .;anÞ and v¼ ðb1;b2;. . .;bnÞ

Theirsum, writtenu_þv, is the vector obtained by adding corresponding components fromuandv. That is,

u_þv¼ ða

1þb1; a2þb2; . . .; anþbnÞ

Theproduct, of the vectoruby a real number k, writtenku, is the vector obtained by multiplying each component ofubyk. That is,

ku_¼k_ða₁;a₂;. . .;a_{nÞ ¼ ð}ka₁;ka₂;. . .;ka_nÞ

Observe that u_þv and ku are also vectors in Rn. The sum of vectors with different numbers of

components is not defined.

Negatives and subtraction are defined inRn as follows:

u_{¼ ð} 1_Þu and u v ¼uþ ð vÞ

The vector uis called thenegativeofu, andu v is called thedifference ofuandv.

Now suppose we are given vectorsu₁;u₂;. . .;u_minRnand scalarsk₁;k₂;. . .;k_minR. We can multiply the vectors by the corresponding scalars and then add the resultant scalar products to form the vector

v¼k

1u1þk2u2þk3u3þ þkmum

Such a vectorv is called alinear combinationof the vectors u

Basic properties of vectors under the operations of vector addition and scalar multiplication are described in the following theorem.

THEOREM1.1: For any vectorsu;v;win Rn and any scalars k;k0 in R,

(i) _ðu_þvÞ þw¼uþ ðvþwÞ, (v) kðuþvÞ ¼kuþkv,

(ii) u_þ0_¼u; (vi) _ðk_þk0_Þu_¼ku_þk0u,

(iii) u_{þ ð} u_{Þ ¼}0; (vii) _ðkk0_Þu_¼k_ðk0u_Þ, (iv) u_þv¼vþu, (viii) 1u¼u.

We postpone the proof of Theorem 1.1 until Chapter 2, where it appears in the context of matrices (Problem 2.3).

Supposeuandvare vectors inRn for whichu¼kvfor some nonzero scalarkinR. Thenuis called a

multipleofv. Also, uis said to be in thesameoropposite directionasv according to whetherk>0 or

k<0.

1.4

Dot (Inner) Product

Consider arbitrary vectorsuandv in Rn; say,

u_{¼ ð}a1;a2;. . .;anÞ and v¼ ðb1;b2;. . .;bnÞ

Thedot productorinner productofuandv is denoted and defined by

uv¼a

1b1þa2b2þ þanbn

That is, uv is obtained by multiplying corresponding components and adding the resulting products.

The vectorsuandv are said to beorthogonal (orperpendicular) if their dot product is zero—that is, if

uv¼0.

EXAMPLE 1.3

(a) Letu¼ ð1; 2;3Þ,v¼ ð4;5; 1Þ,w¼ ð2;7;4Þ. Then, uv¼1ð4Þ 2ð5Þ þ3ð 1Þ ¼4 10 3¼ 9

uw_¼2 14_þ12_¼0; vw¼8þ35 4¼39 Thus,uand ware orthogonal.

(b) Letu¼

2 3 4

2

4 3

5andv¼

3 1 2

2

4 3

5. Thenuv¼6 3þ8¼11.

(c) Supposeu¼ ð1;2;3;4Þandv¼ ð6;k; 8;2Þ. Findkso thatuandvare orthogonal.

First obtainuv¼6þ2k 24þ8¼ 10þ2k. Then setuv¼0 and solve fork:

10_þ2k_¼0 or 2k_¼10 or k_¼5

Basic properties of the dot product inRn (proved in Problem 1.13) follow.

THEOREM1.2: For any vectorsu;v;win Rn and any scalar k inR:

(i) _ðu_þvÞ w¼uwþvw; (iii) uv¼vu,

(ii) _ðku_Þ v¼kðuvÞ, (iv) uu0;anduu¼0 iffu¼0.

Note that (ii) says that we can“takekout”from the first position in an inner product. By (iii) and (ii),

That is, we can also“takek out”from the second position in an inner product.

The space Rn with the above operations of vector addition, scalar multiplication, and dot product is usually calledEuclidean n-space.

is the unique unit vector in the same direction asv. The process of findingv^fromvis callednormalizingv.

EXAMPLE 1.4

Thuswis a unit vector, butvis not a unit vector. However, we can normalizevas follows:

^

This is the unique unit vector in the same direction asv.

The following formula (proved in Problem 1.14) is known as the Schwarz inequality or Cauchy–

Schwarz inequality. It is used in many branches of mathematics.

THEOREM1.3 (Schwarz): For any vectors u;v in Rn,juvj kukkvk.

Using the above inequality, we also prove (Problem 1.15) the following result known as the“triangle inequality”or Minkowski’s inequality.

THEOREM1.4 (Minkowski): For any vectorsu;v inRn,kuþvk kuk þ kvk.

Theangleybetween nonzero vectorsu;v inRn is defined by

cosy_¼ uv ku_kkvk

This definition is well defined, because, by the Schwarz inequality (Theorem 1.3),

1 uv

kukkvk1

Note that if uv¼0, then y¼90 (or y¼p=2). This then agrees with our previous definition of

orthogonality.

Theprojectionof a vectoruonto a nonzero vector v is the vector denoted and defined by

proj_ðu;vÞ ¼ u

We show below that this agrees with the usual notion of vector projection in physics.

EXAMPLE 1.5

(b) Consider the vectorsuandvin Fig. 1-2(a) (with respective endpointsAandB). The (perpendicular) projection of uontovis the vectoru* with magnitude

ku*_{k ¼ k}u_kcosy_{¼ k}u_k uv ku_kvk¼

uv

kvk

To obtainu*, we multiply its magnitude by the unit vector in the direction ofv, obtaining

u*_{¼ k}u*_k v

1.5

Located Vectors, Hyperplanes, Lines, Curves in R

n

This section distinguishes between ann-tupleP_ða_iÞ P_ða₁;a₂;. . .;a_nÞ viewed as a point inRn and an

n-tupleu_{¼ ½}c₁;c₂;. . .;c_nŠ viewed as a vector (arrow) from the originO to the pointC_ðc₁;c₂;. . .;c_nÞ.

Located Vectors

Any pair of pointsA_ða_iÞandB_ðb_iÞinRn defines thelocated vectorordirected line segmentfromAtoB, writtenAB!. We identify AB!with the vector

u_¼B A_{¼ ½}b₁ a₁; b₂ a₂; . . .; b_n a_nŠ

because AB! and u have the same magnitude and direction. This is pictured in Fig. 1-2(b) for the points A_ða₁;a₂;a_3Þ and B_ðb₁;b₂;b_3Þ in R3 and the vector u_¼B A which has the endpoint

P_ðb₁ a₁, b₂ a₂,b₃ a_3Þ.

Hyperplanes

Ahyperplane H inRn is the set of points_ðx₁;x₂;. . .;x_nÞthat satisfy a linear equation

a₁x_1þa₂x_{2þ þ}a_nx_{n ¼}b

where the vectoru_{¼ ½}a₁;a₂;. . .;a_nŠof coefficients is not zero. Thus a hyperplaneHinR2is a line, and a hyperplaneH inR3 is a plane. We show below, as pictured in Fig. 1-3(a) forR3, thatuis orthogonal to any directed line segmentPQ!, whereP_ðp_iÞandQ_ðq_iÞare points inH:[For this reason, we say thatuis

normaltoH and thatH is normaltou:]

BecauseP_ðp_iÞ andQ_ðq_iÞbelong to H;they satisfy the above hyperplane equation—that is,

a₁p_1þa₂p_{2þ þ}a_np_n¼b and a₁q_1þa₂q_{2þ þ}a_nq_{n ¼}b

v ¼PQ!¼Q P¼ ½q

1 p1;q2 p2;. . .;qn pnŠ

Let

Then

uv¼a

1ðq1 p1Þ þa2ðq2 p2Þ þ þanðqn pnÞ

¼ ða₁q_1þa₂q_{2þ þ}a_nq_{nÞ ð}a₁p_1þa₂p_{2þ þ}a_np_{nÞ ¼}b b_¼0

Thusv¼PQ!is orthogonal tou; as claimed.

Figure 1-3

(a) (b)

O

P Q

L P P + t1u

P − t2u u

u

Lines in R

n

The line L in Rn passing through the point P_ðb₁;b₂;. . .;b_nÞ and in the direction of a nonzero vector

u_{¼ ½}a₁;a₂;. . .;a_nŠconsists of the pointsX_ðx₁;x₂;. . .;x_nÞthat satisfy

X_¼P_þtu or

x_1¼a₁t_þb₁ x_2¼a₂t_þb₂ :::::::::::::::::::: x_n¼a_nt_þb_n

or L_ðt_{Þ ¼ ð}a_it_þb_iÞ

8 > > < > > :

where theparameter t takes on all real values. Such a line Lin R3 is pictured in Fig. 1-3(b).

EXAMPLE 1.6

(a) LetH be the plane inR3 corresponding to the linear equation 2x 5yþ7z¼4. Observe that Pð1;1;1_Þ and

Qð5;4;2Þare solutions of the equation. ThusPandQand the directed line segment

v¼PQ!¼Q P¼ ½5 1; 4 1; 2 1Š ¼ ½4;3;1Š

lie on the planeH. The vectoru¼ ½2; 5;7Šis normal toH, and, as expected,

uv¼ ½2; 5;7Š ½4;3;1Š ¼8 15þ7¼0 That is,uis orthogonal tov.

(b) Find an equation of the hyperplaneH inR4 that passes through the pointPð1;3; 4;2Þ and is normal to the vectoru¼ ½4; 2;5;6Š.

The coefficients of the unknowns of an equation ofHare the components of the normal vectoru; hence, the equation ofHmust be of the form

4x1 2x2þ5x3þ6x4¼k

SubstitutingPinto this equation, we obtain

4_ð1_Þ 2_ð3_{Þ þ}5_ð 4_{Þ þ}6_ð2_{Þ ¼}k or 4 6 20_þ12_¼k or k_¼ 10

Thus, 4x₁ 2x_2þ5x_3þ6x_4¼ 10 is the equation ofH.

(c) Find the parametric representation of the lineLinR4passing through the pointPð1;2;3; 4Þand in the direction ofu¼ ½5;6; 7;8Š. Also, find the pointQonLwhent¼1.

Substitution in the above equation forLyields the following parametric representation:

x1 ¼5tþ1; x2¼6tþ2; x3 ¼ 7tþ3; x4 ¼8t 4

or, equivalently,

LðtÞ ¼ ð5tþ1;6tþ2; 7tþ3;8t 4Þ

Note thatt¼0 yields the pointPonL. Substitution oft¼1 yields the pointQð6;8; 4;4ÞonL.

Curves in R

n

LetDbe an interval (finite or infinite) on the real lineR. A continuous functionF:D!Rn is acurvein

Rn. Thus, to each point t₂Dthere is assigned the following point inRn:

F_ðt_{Þ ¼ ½}F_1ðt_Þ;F_2ðt_Þ;. . .;F_nðt_ފ

Moreover, the derivative (if it exists) ofF_ðt_Þ yields the vector

V_ðt_{Þ ¼}dFðtÞ dt ¼

dF1ðtÞ dt ;

dF2ðtÞ dt ;. . .;

dFnðtÞ dt

which is tangent to the curve. NormalizingV_ðt_Þyields

T_ðt_{Þ ¼} VðtÞ kV_ðt_Þk

Thus, T_ðtÞis the unit tangent vector to the curve. (Unit vectors with geometrical significance are often presented in bold type.)

EXAMPLE 1.7 Consider the curveFðtÞ ¼ ½sint;cost;tŠinR3. Taking the derivative ofFðtÞ[or each component of

FðtÞ] yields

V_ðt_{Þ ¼ ½}cost; sint;1_Š

which is a vector tangent to the curve. We normalizeVðtÞ. First we obtain

kV_ðt_Þk2 _¼cos2t_þsin2t_þ1_¼1_þ1_¼2

Then the unit tangent vectionTðtÞ to the curve follows:

T_ðt_{Þ ¼} VðtÞ kV_ðt_Þk¼

cost

ffiffiffi 2

p ; sinffiffiffit 2

p ; 1ffiffiffi 2

p

1.6

Vectors in R

3

(Spatial Vectors), ijk Notation

Vectors inR3, calledspatial vectors, appear in many applications, especially in physics. In fact, a special notation is frequently used for such vectors as follows:

i_{¼ ½}1;0;0_Šdenotes the unit vector in thexdirection:

j_{¼ ½}0;1;0_Šdenotes the unit vector in theydirection:

k_{¼ ½}0;0;1_Šdenotes the unit vector in thezdirection:

Then any vectoru_{¼ ½}a;b;c_Šin R3 can be expressed uniquely in the form

u_{¼ ½}a;b;c_{Š ¼}ai_þbj_þck

Because the vectors i;j;k are unit vectors and are mutually orthogonal, we obtain the following dot products:

ii_¼1; jj_¼1; kk_¼1 and ij_¼0; ik_¼0; jk_¼0

Furthermore, the vector operations discussed above may be expressed in the ijk notation as follows. Suppose

u_¼a₁i_þa₂j_þa₃k and v¼b

1iþb2jþb3k

Then

u_þv¼ ða

1þb1Þiþ ða2þb2Þjþ ða3þb3Þk and cu¼ca1iþca2jþca3k

wherecis a scalar. Also,

uv¼a

1b1þa2b2þa3b3 and kuk ¼ ffiffiffiffiffiffiffiffiffi

uu p

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

a2

1þa22þa23 q

EXAMPLE 1.8 Supposeu¼3iþ5j 2kandv¼4i 8jþ7k.

(a) To finduþv, add corresponding components, obtaininguþv¼7i 3jþ5k

(b) To find 3u 2v, first multiply by the scalars and then add:

(c) To finduv, multiply corresponding components and then add: uv¼12 40 14¼ 42

(d) To findkuk, take the square root of the sum of the squares of the components:

ku_{k ¼}pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi9_þ25_þ4_¼pffiffiffiffiffi38

Cross Product

There is a special operation for vectorsuandvinR3that is not defined inRnforn6¼3. This operation is

called thecross productand is denoted byuv. One way to easily remember the formula foruvis to

use the determinant (of order two) and its negative, which are denoted and defined as follows:

a b

Here a and d are called the diagonal elements and b and c are the nondiagonal elements. Thus, the determinant is the productadof the diagonal elements minus the productbcof the nondiagonal elements, but vice versa for the negative of the determinant.

Now supposeu_¼a₁i_þa₂j_þa₃kandv¼b That is, the three components ofuv are obtained from the array

a₁ a₂ a₃ b₁ b₂ b₃

(which contain the components ofuabove the component ofv) as follows:

(1) Cover the first column and take the determinant.

(2) Cover the second column and take the negative of the determinant. (3) Cover the third column and take the determinant.

Note that uv is a vector; hence, uv is also called the vector product or outer product of u

andv.

Remark: The cross products of the vectors i;j;kare as follows:

ij_¼k; jk_¼i; ki_¼j

ji_¼ k; kj_¼ i; ik_¼ j

Thus, if we view the triple_ði;j;k_Þas a cyclic permutation, whereifollowskand hencekprecedesi, then the product of two of them in the given direction is the third one, but the product of two of them in the opposite direction is the negative of the third one.

THEOREM1.5: Letu;v;w be vectors inR3.

(a) The vectoruv is orthogonal to bothuandv.

(b) The absolute value of the _“triple product_”

uvw

represents the volume of the parallelepiped formed by the vectors u;v, w.

[See Fig. 1-4(a).]

We note that the vectorsu;v,uvform a right-handed system, and that the following formula gives

the magnitude ofuv:

kuvk ¼ kukkvksiny

whereyis the angle between uandv.

1.7

Complex Numbers

The set of complex numbers is denoted byC. Formally, a complex number is an ordered pair_ða;b_Þof real numbers where equality, addition, and multiplication are defined as follows:

ða;b_{Þ ¼ ð}c;d_Þ if and only ifa_¼candb_¼d ða;b_{Þ þ ð}c;d_{Þ ¼ ð}a_þc; b_þd_Þ

ða;b_{Þ ð}c;d_{Þ ¼ ð}ac bd; ad_þbc_Þ

We identify the real numberawith the complex number _ða;0_Þ; that is,

a_{$ ð}a;0_Þ

This is possible because the operations of addition and multiplication of real numbers are preserved under the correspondence; that is,

ða;0_{Þ þ ð}b;0_{Þ ¼ ð}a_þb; 0_Þ and _ða;0_{Þ ð}b;0_{Þ ¼ ð}ab;0_Þ

Thus we viewRas a subset of C, and replace _ða;0_Þ byawhenever convenient and possible.

We note that the setCof complex numbers with the above operations of addition and multiplication is a

fieldof numbers, like the setR of real numbers and the setQof rational numbers.

Figure 1-4

(a) (b)

O

Volume = _u._v × _{w Complex plane}

y

x

z

v

O b

a z = _a + _bi

|z| u

The complex number_ð0;1_Þis denoted byi. It has the important property that

i2_¼ii¼ ð0;1_Þð0;1_{Þ ¼ ð} 1;0_{Þ ¼} 1 or i¼pffiffiffiffiffiffiffi1

Accordingly, any complex numberz_{¼ ð}a;b_Þ can be written in the form

z_{¼ ð}a;b_{Þ ¼ ð}a;0_{Þ þ ð}0;b_{Þ ¼ ð}a;0_{Þ þ ð}b;0_{Þ ð}0;1_{Þ ¼}a_þbi

The above notation z_¼a_þbi, whereaRez andbImz are called, respectively, the real and

imaginary partsofz, is more convenient than _ða;b_Þ. In fact, the sum and product of complex numbers

z_¼a_þbi and w_¼c_þdi can be derived by simply using the commutative and distributive laws and

i2 _¼ 1:

z_þw_{¼ ð}a_þbi_{Þ þ ð}c_þdi_{Þ ¼}a_þc_þbi_þdi_{¼ ð}a_þb_{Þ þ ð}c_þd_Þi

zw_{¼ ð}a_þbi_Þðc_þdi_{Þ ¼}ac_þbci_þadi_þbdi2 _{¼ ð}ac bd_{Þ þ ð}bc_þad_Þi

We also define thenegativeofzand subtraction inC by

z_¼ 1z and w z_¼w_{þ ð} z_Þ

Warning:The letteri representingpffiffiffiffiffiffiffi1has no relationship whatsoever to the vectori_{¼ ½}1;0;0_Šin Section 1.6.

Complex Conjugate, Absolute Value

Consider a complex numberz_¼a_þbi. Theconjugate ofzis denoted and defined by

z_¼a_þbi_¼a bi

Thenzz_{¼ ð}a_þbi_Þða bi_{Þ ¼}a2 _b2_i2 _¼a2_þb2_{. Note thatz is real if and only ifz¼z.}

Theabsolute valueofz, denoted by _jz_j, is defined to be the nonnegative square root ofzz. Namely,

jzj ¼pffiffiffiffizz¼pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2_þb2

Note that_jz_jis equal to the norm of the vector_ða;b_Þin R2.

Supposez_6¼0. Then the inversez 1 ofzand division in Cofw byzare given, respectively, by

z 1 _¼ z zz¼

a a2_þb2

b

a2_þb2i and w

z wz

zz ¼wz 1

EXAMPLE 1.10 Supposez¼2þ3iandw¼5 2i. Then

zþw¼ ð2þ3iÞ þ ð5 2iÞ ¼2þ5þ3i 2i¼7þi

zw¼ ð2_þ3iÞð5 2iÞ ¼10_þ15i 4i 6i2¼16_þ11i z¼2þ3i¼2 3i and w¼5 2i¼5þ2i w

z ¼

5 2i

2þ3i¼

ð5 2iÞð2 3iÞ ð2þ3iÞð2 3iÞ¼

4 19i

13 ¼

4 13

19 13i

jzj ¼pffiffiffiffiffiffiffiffiffiffiffi4þ9¼pffiffiffiffiffi13 and jwj ¼pffiffiffiffiffiffiffiffiffiffiffiffiffiffi25þ4¼pffiffiffiffiffi29

Complex Plane

Recall that the real numbersRcan be represented by points on a line. Analogously, the complex numbers

1.8

Vectors in C

n

The set of alln-tuples of complex numbers, denoted byCn, is calledcomplexn-space. Just as in the real case, the elements ofCn are called points or vectors, the elements of C are called scalars, and vector addition inCn and scalar multiplication onCn are given by

½z₁;z₂;. . .;z_{nŠ þ ½}w₁;w₂;. . .;w_{nŠ ¼ ½}z_1þw₁; z_2þw₂; . . .; z_nþw_nŠ z_½z₁;z₂;. . .;z_{nŠ ¼ ½}zz₁;zz₂;. . .;zz_nŠ

where thez_i,w_i, andzbelong toC.

EXAMPLE 1.11 Consider vectorsu¼ ½2þ3i; 4 i; 3Šandv¼ ½3 2i; 5i; 4 6iŠinC3. Then u_þv ¼ ½2þ3i; 4 i; 3Š þ ½3 2i; 5i; 4 6iŠ ¼ ½5þi; 4þ4i; 7 6iŠ

ð5 2i_Þu _{¼ ½ð}5 2i_Þð2_þ3i_Þ; ð5 2i_Þð4 i_Þ; ð5 2i_Þð3_{ފ ¼ ½}16_þ11i; 18 13i; 15 6i_Š

Dot (Inner) Product in C

n

Consider vectorsu_{¼ ½}z1;z2;. . .;znŠandv¼ ½w1;w2;. . .;wnŠinCn. Thedotorinner productofuandvis

denoted and defined by

uv¼z

1w1þz2w2þ þznwn

This definition reduces to the real case becausew_i¼w_i whenw_i is real. The norm of uis defined by

kuk ¼pffiffiffiffiffiffiffiffiffiuu¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiz1z1þz2z2þ þznzn p

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jz1j

2 þ jz2j

2

þ þ jv

nj 2 q

We emphasize thatuuand so_ku_kare real and positive when u_6¼0 and 0 whenu_¼0.

EXAMPLE 1.12 Consider vectorsu¼ ½2þ3i; 4 i; 3þ5iŠandv¼ ½3 4i; 5i; 4 2iŠinC3. Then uv¼ ð2þ3iÞð3 4iÞ þ ð4 iÞð5iÞ þ ð3þ5iÞð4 2iÞ

¼ ð2_þ3i_Þð3_þ4i_{Þ þ ð}4 i_Þð 5i_{Þ þ ð}3_þ5i_Þð4_þ2i_Þ ¼ ð 6_þ13i_{Þ þ ð} 5 20i_{Þ þ ð}2_þ26i_{Þ ¼} 9_þ19i

uu_{¼ j}2_þ3i_j2_{þ j}4 i_j2_{þ j}3_þ5i_j2 _¼ 4_þ9_þ16_þ1_þ9_þ25 _¼ 64

ku_{k ¼}pffiffiffiffiffi64_¼8

The spaceCn with the above operations of vector addition, scalar multiplication, and dot product, is calledcomplex Euclideann-space. Theorem 1.2 for Rn also holds forCn if we replaceuv ¼vuby

uv¼uv

On the other hand, the Schwarz inequality (Theorem 1.3) and Minkowski_’s inequality (Theorem 1.4) are true forCn with no changes.

SOLVED PROBLEMS

Vectors in Rn

1.1. Determine which of the following vectors are equal:

u_{1 ¼ ð}1;2;3_Þ; u_{2¼ ð}2;3;1_Þ; u_{3¼ ð}1;3;2_Þ; u_{4¼ ð}2;3;1_Þ

1.2. Letu_{¼ ð}2; 7;1_Þ, v¼ ð 3;0;4Þ,w¼ ð0;5; 8Þ. Find:

(a) 3u 4v,

(b) 2u_þ3v 5w.

First perform the scalar multiplication and then the vector addition.

(a) 3u 4v¼3ð2; 7;1Þ 4ð 3;0;4Þ ¼ ð6; 21;3Þ þ ð12;0; 16Þ ¼ ð18; 21; 13Þ

First perform the scalar multiplication and then the vector addition:

(a) 5u 2v¼5

(a) Because the vectors are equal, set the corresponding entries equal to each other, yielding

x¼2; 3¼xþy

(It is more convenient to write vectors as columns than as rows when forming linear combinations.) Set corresponding entries equal to each other to obtain

1.6. Writev¼ ð2; 5;3Þas a linear combination of

u1¼ ð1; 3;2Þ;u2¼ ð2; 4; 1Þ;u3¼ ð1; 5;7Þ:

Find the equivalent system of linear equations and then solve. First,

2 5 3

2 4

3 5_¼x

1 3 2

2 4

3 5_þy

2 4 1

2 4

3 5_þz

1 5 7

2 4

3

5_¼ 3xxþ24yyþ5zz

2x yþ7z 2

4

3 5

Set the corresponding entries equal to each other to obtain

xþ2yþ z¼ 2 3x 4y 5z¼ 5 2x yþ7z¼ 3

or

xþ2yþ z¼ 2 2y 2z¼ 1 5yþ5z¼ 1

or

xþ2yþ z¼2 2y 2z¼1 0¼3

The third equation, 0xþ0yþ0z¼3, indicates that the system has no solution. Thus,vcannot be written as a

linear combination of the vectorsu1,u2,u3.

Dot (Inner) Product, Orthogonality, Norm in Rn

1.7. Finduv where:

(a) u¼ ð2; 5;6Þ andv¼ ð8;2; 3Þ,

(b) u¼ ð4;2; 3;5; 1Þandv¼ ð2;6; 1; 4;8Þ.

Multiply the corresponding components and add:

(a) uv¼2ð8Þ 5ð2Þ þ6ð 3Þ ¼16 10 18¼ 12

(b) uv¼8þ12þ3 20 8¼ 5

1.8. Letu_{¼ ð}5;4;1_Þ, v¼ ð3; 4;1Þ,w¼ ð1; 2;3Þ. Which pair of vectors, if any, are perpendicular

(orthogonal)?

Find the dot product of each pair of vectors:

uv¼15 16þ1¼0; vw¼3þ8þ3¼14; uw¼5 8þ3¼0

Thus,uandvare orthogonal,uandware orthogonal, butvandware not.

1.9. Findk so that uandv are orthogonal, where: (a) u¼ ð1;k; 3Þandv¼ ð2; 5;4Þ,

(b) u¼ ð2;3k; 4;1;5Þandv¼ ð6; 1;3;7;2kÞ.

Computeuv, setuvequal to 0, and then solve fork:

(a) uv¼1ð2Þ þkð 5Þ 3ð4Þ ¼ 5k 10. Then 5k 10¼0, ork¼ 2.

(b) uv¼12 3k 12þ7þ10k¼7kþ7. Then 7kþ7¼0, ork¼ 1.

1.10. Find_ku_k, where: (a) u_{¼ ð}3; 12; 4_Þ, (b) u_{¼ ð}2; 3;8; 7_Þ.

First findkuk2¼uuby squaring the entries and adding. Thenkuk ¼ ffiffiffiffiffiffiffiffiffiffi kuk2 q

.

1.11. Recall thatnormalizing a nonzero vector v means finding the unique unit vector v^in the same

(c) Note thatwand any positive multiple ofwwill have the same normalized form. Hence, first multiplywby 12 to“clear fractions”—that is, first findw0_{¼12w¼ ð6;8; 3Þ. Then}

i is nonnegative for eachi, and because the sum of nonnegative real numbers is nonnegative,

1.14. Prove Theorem 1.3 (Schwarz):_juvj kukkvk. For any real numbert, and using Theorem 1.2, we have

0 ðtuþvÞ ðtuþvÞ ¼t2ðuuÞ þ2tðuvÞ þ ðvvÞ ¼ kuk2t2þ2ðuvÞtþ kvk2

Let a¼ kuk2, b¼2ðuvÞ,c¼ kvk2. Then, for every value of t, at2þbtþc0. This means that the

quadratic polynomial cannot have two real roots. This implies that the discriminantD¼b2 _{4ac0 or,}

equivalently,b2_{4ac. Thus,}

4ðuvÞ24kuk2kvk2

Dividing by 4 gives us our result.

1.15. Prove Theorem 1.4 (Minkowski):_ku_þvk kuk þ kvk. By the Schwarz inequality and other properties of the dot product,

kuþvk2¼ ðuþvÞ ðuþvÞ ¼ ðuuÞ þ2ðuvÞ þ ðvvÞ kuk2þ2kukkvk þ kvk2¼ ðkuk þ kvkÞ2

Taking the square root of both sides yields the desired inequality.

Points, Lines, Hyperplanes in Rn

Here we distinguish between an n-tuple P_ða₁;a₂;. . .;a_nÞ viewed as a point in Rn and an n-tuple

u_{¼ ½}c₁;c₂;. . .;c_nŠ viewed as a vector (arrow) from the originOto the point C_ðc₁;c₂;. . .;c_nÞ.

1.16. Find the vectoruidentified with the directed line segment PQ!for the points:

(a) P_ð1; 2;4_ÞandQ_ð6;1; 5_ÞinR3, (b) P_ð2;3; 6;5_Þ andQ_ð7;1;4; 8_ÞinR4.

(a) u¼PQ!¼Q P¼ ½6 1; 1 ð 2Þ; 5 4Š ¼ ½5;3; 9Š

(b) u¼PQ!¼Q P¼ ½7 2; 1 3; 4þ6; 8 5Š ¼ ½5; 2;10; 13Š

1.17. Find an equation of the hyperplaneH inR4 that passes throughP_ð3; 4;1; 2_Þand is normal to

u_{¼ ½}2;5; 6; 3_Š.

The coefficients of the unknowns of an equation ofHare the components of the normal vectoru. Thus, an equation ofH is of the form 2x₁þ5x₂ 6x₃ 3x₄¼k. SubstitutePinto this equation to obtaink¼ 26. Thus, an equation ofHis 2x1þ5x2 6x3 3x4¼ 26.

1.18. Find an equation of the planeH inR3 that contains P_ð1; 3; 4_Þand is parallel to the planeH0

determined by the equation 3x 6y_þ5z_¼2.

The planesH andH0are parallel if and only if their normal directions are parallel or antiparallel (opposite direction). Hence, an equation ofHis of the form 3x 6yþ5z¼k. SubstitutePinto this equation to obtain

k¼1. Then an equation ofHis 3x 6yþ5z¼1.

1.19. Find a parametric representation of the lineLinR4 passing throughP_ð4; 2;3;1_Þin the direction ofu_{¼ ½}2;5; 7;8_Š.

HereLconsists of the pointsXðxiÞthat satisfy

X¼Pþtu or xi¼aitþbi or LðtÞ ¼ ðaitþbiÞ where the parameterttakes on all real values. Thus we obtain

1.20. LetC be the curveF_ðt_{Þ ¼ ð}t2; 3t 2; t3; t2_þ5_ÞinR4, where 0t4.

(a) Find the pointPonC corresponding tot_¼2. (b) Find the initial pointQand terminal point Q0 ofC. (c) Find the unit tangent vectorTto the curve C whent_¼2.

(a) Substitutet_¼2 intoF_ðt_Þto getP_¼f_ð2_{Þ ¼ ð}4;4;8;9_Þ.

(b) The parameter t ranges from t¼0 to t¼4. Hence, Q¼fð0Þ ¼ ð0; 2;0;5Þ and

Q0¼Fð4Þ ¼ ð16;10;64;21Þ.

(c) Take the derivative ofFðtÞ—that is, of each component ofFðtÞ—to obtain a vectorVthat is tangent to the curve:

VðtÞ ¼dFðtÞ

dt ¼ ½2t;3;3t

2_;2t Š

Now find V when t¼2; that is, substitute t¼2 in the equation for VðtÞ to obtain

V¼Vð2Þ ¼ ½4;3;12;4Š. Then normalizeV to obtain the desired unit tangent vectorT. We have

kVk ¼pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi16_þ9_þ144_þ16_¼pffiffiffiffiffiffiffiffi185 and T_¼ ffiffiffiffiffiffiffiffi4

185

p ; ffiffiffiffiffiffiffiffi3

185

p ; 12ffiffiffiffiffiffiffiffi

185

p ; ffiffiffiffiffiffiffiffi4

185

p

Spatial Vectors (Vectors in R3_{),ijk Notation, Cross Product}

1.21. Letu_¼2i 3j_þ4k,v¼3iþj 2k,w¼iþ5jþ3k. Find:

(a) u_þv, (b) 2u 3vþ4w, (c) uv anduw, (d) kukandkvk. Treat the coefficients ofi,j,kjust like the components of a vector in R3.

(a) Add corresponding coefficients to getuþv¼5i 2j 2k.

(b) First perform the scalar multiplication and then the vector addition:

2u 3vþ4w¼ ð4i 6jþ8kÞ þ ð 9i 3jþ6kÞ þ ð4iþ20jþ12kÞ ¼ iþ11jþ26k

(c) Multiply corresponding coefficients and then add:

uv¼6 3 8¼ 5 and uw¼2 15þ12¼ 1

(d) The norm is the square root of the sum of the squares of the coefficients:

ku_{k ¼}p4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi_þ9_þ16_¼pffiffiffiffiffi29 and _kvk ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

9þ1þ4

p

¼pffiffiffiffiffi14

1.22. Find the (parametric) equation of the lineL:

(a) through the pointsP_ð1;3;2_Þ andQ_ð2;5; 6_Þ;

(b) containing the point P_ð1; 2;4_Þ and perpendicular to the plane H given by the equation 3x_þ5y_þ7z_¼15:

(a) First findv¼PQ!¼Q P¼ ½1;2; 8Š ¼iþ2j 8k. Then

L_ðt_{Þ ¼ ð}t_þ1; 2t_þ3; 8t_þ2_{Þ ¼ ð}t_þ1_Þi_{þ ð}2t_þ3_Þj_{þ ð} 8t_þ2_Þk

(b) BecauseLis perpendicular toH, the lineLis in the same direction as the normal vectorN¼3iþ5jþ7k

toH. Thus,

LðtÞ ¼ ð3tþ1; 5t 2; 7tþ4Þ ¼ ð3tþ1Þiþ ð5t 2Þjþ ð7tþ4Þk

1.23. LetSbe the surfacexy2_þ2yz_¼16 inR3.

(a) The formula for the normal vector to a surfaceF_ðx;y;z_{Þ ¼}0 is

Nðx;y;zÞ ¼FxiþFyjþFzk

whereF_x,F_y,F_z are the partial derivatives. UsingFðx;y;zÞ ¼xy2_{þ2yz 16, we obtain}

F_x¼y2; F_y¼2xy_þ2z; F_z¼2y

Thus,Nðx;y;zÞ ¼y2_{iþ ð2xyþ2zÞjþ2yk.}

(b) The normal to the surfaceSat the pointPis

NðPÞ ¼Nð1;2;3Þ ¼4iþ10jþ4k

Hence,N¼2iþ5jþ2kis also normal toSatP. Thus an equation ofHhas the form 2xþ5yþ2z¼c. SubstitutePin this equation to obtainc¼18. Thus the tangent planeHtoSatPis 2xþ5yþ2z¼18.

1.24. Evaluate the following determinants and negative of determinants of order two:

(a) We have

u ðuvÞ ¼a₁ða₂b₃ a₃b₂Þ þa₂ða₃b₁ a₁b₃Þ þa₃ða₁b₂ a₂b₁Þ ¼a1a2b3 a1a3b2þa2a3b1 a1a2b3þa1a3b2 a2a3b1¼0

Thus,uvis orthogonal tou. Similarly,uvis orthogonal tov.

(b) We have

kuvk2¼ ða₂b₃ a₃b₂Þ2þ ða₃b₁ a₁b₃Þ2þ ða₁b₂ a₂b₁Þ2 ð1Þ ðuuÞðvvÞ ðuvÞ2¼ ða2

1þa22þa23Þðb21þb22þb23Þ ða1b1þa2b2þa3b3Þ 2

ð2Þ

Expansion of the right-hand sides of (1) and (2) establishes the identity.

Complex Numbers, Vectors in Cn

1.29. Supposez_¼5_þ3i andw_¼2 4i. Find: (a) z_þw, (b) z w, (c) zw.

Use the ordinary rules of algebra together withi2_{¼ 1 to obtain a result in the standard formaþbi.}

(a) z_þw_{¼ ð}5_þ3i_{Þ þ ð}2 4i_{Þ ¼}7 i

(b) z w¼ ð5þ3iÞ ð2 4iÞ ¼5þ3i 2þ4i¼3þ7i

(c) zw¼ ð5þ3iÞð2 4iÞ ¼10 14i 12i2_{¼10 14iþ12¼22 14i}

1.30. Simplify: (a) _ð5_þ3i_Þð2 7i_Þ, (b) _ð4 3i_Þ2, (c) _ð1_þ2i_Þ3.

(a) ð5þ3iÞð2 7iÞ ¼10þ6i 35i 21i2_{¼31 29i}

(b) ð4 3iÞ2¼16 24iþ9i2_{¼7 24i}

(c) ð1þ2iÞ3¼1þ6iþ12i2_þ8i3_{¼1þ6i 12 8i¼ 11 2i}

1.31. Simplify: (a) i0;i3;i4, (b) i5;i6;i7;i8, (c) i39; i174,i252,i317:

(a) i0_{¼1, i}3_¼i2_{ðiÞ ¼ ð 1ÞðiÞ ¼ i; i}4_{¼ ði}2_Þði2_{Þ ¼ ð 1Þð 1Þ ¼1}

(b) i5¼ ði4ÞðiÞ ¼ ð1ÞðiÞ ¼i, i6¼ ði4Þði2Þ ¼ ð1Þði2Þ ¼i2¼ 1, i7¼i3¼ i, i8¼i4¼1

(c) Usingi4¼1 andin¼i4qþr¼ ði4Þqir¼1qir¼ir, divide the exponentnby 4 to obtain the remainderr:

i39¼i4ð9Þþ3¼ ði4Þ9i3¼19i3¼i3¼ i; i174¼i2¼ 1; i252¼i0¼1; i317¼i1¼i

1.32. Find the complex conjugate of each of the following:

(a) 6_þ4i, 7 5i, 4_þi, 3 i, (b) 6, 3, 4i, 9i.

(a) 6þ4i¼6 4i, 7 5i¼7þ5i, 4þi¼4 i, 3 i¼ 3þi

(b) 6¼6, 3¼ 3, 4i¼ 4i, 9i¼9i

(Note that the conjugate of a real number is the original number, but the conjugate of a pure imaginary number is the negative of the original number.)

1.33. Findzz and_jz_jwhenz_¼3_þ4i.

Forz¼aþbi, usezz¼a2þb2and z¼pffiffiffiffizz¼pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2_þb2_.

zz¼9þ16¼25; jzj ¼pffiffiffiffiffi25¼5

1.34. Simpify2 7i 5_þ3i:

To simplify a fraction z=w of complex numbers, multiply both numerator and denominator by w, the conjugate of the denominator:

2 7i

5_þ3i¼

ð2 7iÞð5 3iÞ ð5_þ3iÞð5 3iÞ¼

11 41i

34 ¼

11 34

1.35. Prove: For any complex numbersz, w₂C, (i)z_þw_¼z_þw, (ii)zw_¼zw, (iii)z_¼z.

Supposez¼aþbiand w¼cþdiwherea;b;c;d2R. (i) zþw¼ ðaþbiÞ þ ðcþdiÞ ¼ ðaþcÞ þ ðbþdÞi

¼ ðaþcÞ ðbþdÞi ¼ aþc bi di ¼ ða biÞ þ ðc diÞ ¼ zþw

(ii) zw_{¼ ð}a_þbi_Þðc_þdi_{Þ ¼ ð}ac bd_{Þ þ ð}ad_þbc_Þi ¼ ðac bdÞ ðadþbcÞi ¼ ða biÞðc diÞ ¼ zw

(iii) z¼aþbi¼a bi¼a ð bÞi¼aþbi¼z

1.36. Prove: For any complex numbersz;w₂C,_jzw_{j ¼ j}z_jjw_j.

By (ii) of Problem 1.35,

jzw_j2 _{¼ ð}zw_Þðzw_{Þ ¼ ð}zw_Þðzw_{Þ ¼ ð}zz_Þðww_{Þ ¼ j}z_j2_jw_j2

The square root of both sides gives us the desired result.

1.37. Prove: For any complex numbersz;w₂C,_jz_þw_{j j}z_{j þ j}w_j.

Supposez¼aþbiand w¼cþdiwherea;b;c;d2R. Consider the vectorsu¼ ða;bÞ andv¼ ðc;dÞin R2. Note that

jzj ¼pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2_þb2_{¼ kuk; jwj ¼}pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi_c2_þd2_{¼ k}vk

and

jzþwj ¼ jðaþcÞ þ ðbþdÞij ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðaþcÞ2þ ðbþdÞ2 q

¼ kðaþc;bþdÞk ¼ kuþvk

By Minkowski’s inequality (Problem 1.15),kuþvk kuk þ kvk, and so jzþwj ¼ kuþvk kuk þ kvk ¼ jzj þ jwj

1.38. Find the dot products uv and vu where: (a) u¼ ð1 2i; 3þiÞ, v ¼ ð4þ2i; 5 6iÞ;

(b) u_{¼ ð}3 2i; 4i; 1_þ6i_Þ,v¼ ð5þi; 2 3i; 7þ2iÞ. Recall that conjugates of the second vector appear in the dot product

ðz1;. . .;znÞ ðw1;. . .;wnÞ ¼z1w1þ þznwn (a) uv¼ ð1 2iÞð4þ2iÞ þ ð3þiÞð5 6iÞ

¼ ð1 2iÞð4 2iÞ þ ð3þiÞð5þ6iÞ ¼ 10iþ9þ23i ¼ 9þ13i vu¼ ð4þ2iÞð1 2iÞ þ ð5 6iÞð3þiÞ

¼ ð4þ2iÞð1þ2iÞ þ ð5 6iÞð3 iÞ ¼ 10iþ9 23i ¼ 9 13i

(b) uv¼ ð3 2iÞð5þiÞ þ ð4iÞð2 3iÞ þ ð1þ6iÞð7þ2iÞ

¼ ð3 2iÞð5 iÞ þ ð4iÞð2þ3iÞ þ ð1þ6iÞð7 2iÞ ¼ 20þ35i vu¼ ð5þiÞð3 2iÞ þ ð2 3iÞð4iÞ þ ð7þ2iÞð1þ6iÞ

¼ ð5þiÞð3þ2iÞ þ ð2 3iÞð 4iÞ þ ð7þ2iÞð1 6iÞ ¼ 20 35i

In both cases,vu¼uv. This holds true in general, as seen in Problem 1.40.

1.39. Letu_{¼ ð}7 2i; 2_þ5i_Þandv¼ ð1þi; 3 6iÞ. Find:

(a) u_þv, (b) 2iu, (c) ð3 iÞv, (d) uv, (e) kukandkvk. (a) uþv¼ ð7 2iþ1þi; 2þ5i 3 6iÞ ¼ ð8 i; 1 iÞ

(b) 2iu¼ ð14i 4i2_{; 4iþ10i}2_{Þ ¼ ð4þ14i; 10þ4iÞ}