Intermediate dynamics a linear algebraic approach

1.1.2 Subspaces of a Vector Space 12 1.2 The Basis of a Vector Space 13 1.2.1 Spanning Sets 14 1.2.2 Linear Independence 16 A Test for Linear Independence of n-tuples: Reduction to Echel

Frederick F Ling

Editor-in-Chief

Mechanical Engineering Series

J Angeles, Fundamentals of Robotic Mechanical Systems:

Theory, Methods, and Algorithms, 2nd ed

P Basu, C Kefa, and L Jestin, Boilers and Burners: Design and Theory

J.M Berthelot, Composite Materials:

Mechanical Behavior and Structural Analysis

I.J Busch-Vishniac, Electromechanical Sensors and Actuators

J Chakrabarty, Applied Plasticity

K.K Choi and N.H Kim, Structural Sensitivity Analysis and Optimization 1: Linear Systems

K.K Choi and N.H Kim, Structural Sensitivity Analysis and Optimization 2: Nonlinear Systems and Applications

G Chiyssolouris, Laser Machining: Theory and Practice

V.N Constantinescu, Laminar Viscous Flow

G.A Costello, Theory of Wire Rope, 2nd Ed

K Czolczynski, Rotordynamics of Gas-Lubricated Journal Bearing Systems M.S Darlow, Balancing of High-Speed Machinery

J.F Doyle, Nonlinear Analysis of Thin-Walled Structures: Statics,

Dynamics, and Stability

J.F Doyle, Wave Propagation in Structures:

Spectral Analysis Using Fast Discrete Fourier Transforms, 2nd ed

PA Engel, Structural Analysis of Printed Circuit Board Systems

AC Fischer-Cripps, Introduction to Contact Mechanics

A.C Fischer-Cripps, Nanoindentations, 2nd ed

J Garcia de Jalon and E Bayo, Kinematic and Dynamic Simulation of

Multibody Systems: The Real-Time Challenge

W.K Gawronski, Advanced Structural Dynamics and Active Control of Structures

W.K Gawronski, Dynamics and Control of Structures: A Modal Approach

G Genta, Dynamics of Rotating Systems

(continued after index)

R A Rowland

Intermediate Dynamics:

A Linear Algebraic Approach

^ Spri mger

R A Howland

University of Notre Dame

Editor-in-Chief

Frederick F Ling

Earnest F Gloyna Regents Chair Emeritus in Engineering

Department of Mechanical Engineering

The University of Texas at Austin

Austin, TX 78712-1063, USA

and

Distinguished William Howard Hart

Professor Emeritus

Department of Mechanical Engineering,

Aeronautical Engineering and Mechanics

Rensselaer Polytechnic Institute

Troy, NY 12180-3590, USA

Intermediate Dynamics: A Linear Algebraic Approach

ISBN 0-387-28059-6 e-ISBN 0-387-28316-1 Printed on acid-free paper ISBN 978-0387-28059-2

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The Mechanical Engineering Series features graduate texts and research monographs to address the need for information in contemporary mechanical engineering, including areas of concentration of applied mechanics, biomechanics, computational mechanics, dynamical systems and control, energetics, mechanics of materials, processing, pro-duction systems, thermal science, and tribology

Advisory Board/Series Editors

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Series Preface

Mechanical engineering, and engineering discipline bom of the needs of the dustrial revolution, is once again asked to do its substantial share in the call for industrial renewal The general call is urgent as we face profound issues of pro-ductivity and competitiveness that require engineering solutions, among others The Mechanical Engineering Series is a series featuring graduate texts and re-search monographs intended to address the need for information in contemporary areas of mechanical engineering

in-The series is conceived as a comprehensive one that covers a broad range of concentrations important to mechanical engineering graduate education and re-search We are fortunate to have a distinguished roster of consulting editors, each

an expert in one of the areas of concentration The names of the consulting editors are listed on page vi of this volume The areas of concentration are applied me-chanics, biomechanics, computational mechanics, dynamic systems and control, energetics, mechanics of materials, processing, thermal science, and tribology

a small number of texts available for such offerings, there is a notable paucity aimed at "mainstream" undergraduates, and instructors often resort to utiliz-ing sections in the first Mechanics text not covered in the introductory course,

at least for the 3-D rigid body dynamics Though closely allied to its planar counterpart, this topic is far more complex than its predecessor: in kinematics, one must account for possible change in direction of the angular velocity; and the kinetic "moment of inertia," a simple scalar in the planar formulation, must

be replaced by a tensor quantity If elementary texts' presentation of planar dynamics is adequate, their treatment of three-dimensional dynamics is rather

less satisfactory: It is common to expand vector equations of motion in

compo-nents—in a particular choice of axes—and consider only a few special instances

of their application {e,g, fixed-axis rotation in the Euler equations of motion) The presentation of principal coordinates is typically somewhat ad hoc, either

merely stating the procedure to find such axes in general, or even more monly invoking the "It can be shown t h a t " mantra Machines seem not to

com-exist in 3-D! And equations of motion for the gyroscope are derived

indepen-dently of the more general ones—a practice lending a certain air of mystery to this important topic

Such an approach can be frustrating to the student with any degree of ity and is counterproductive pedagogically: the component-wise expression of vector quantities has long since disappeared from even Sophomore-level courses

curios-in Mechanics, curios-in good part because the complexity of notation obscures the relative simplicity of the concepts involved But the Euler equations can be expressed both succinctly and generally through the introduction of matrices The typical exposition of principal axes overlooks the fact that this is precisely

the same device used to find the same "principal axes" in solid mechanics

(ex-plicitly through a rotation); few students recognize this fact, and, unfortunately, few instructors take the opportunity to point this out and unify the concepts And principal axes themselves are, in fact, merely an application of an even more general technique utilized in linear algebra leading to the diagonalization

of matrices (at least the real, symmetric ones encountered in both solid ics and dynamics) These facts alone suggest a linear algebraic approach to the subject

mechan-A knowledge of linear algebra is, however, more beneficial to the scientist and engineer than merely to be able to diagonalize matrices: Eigenvectors and

eigenvalues pervade both fields; yet, while students can typically find these

quantities and use them to whatever end they have been instructed in, few can answer the simple question "What is an eigenvector?" As the field of robotics becomes ever more mainstream, a facility with [3-D] rotation matrices becomes increasingly important Even the mundane issue of solving linear equations is often incomplete or, worse still, inaccurate: "All you need is as many equations

as unknowns If you have fewer than that, there is no solution." (The first of these statements is incomplete, the second downright wrong!) Such fallacies are likely not altogether the students' fault: few curricula allow the time to devote

a full, formal course to the field, and knowledge of the material is typically gleaned piecemeal on an "as-need" basis The result is a fractionated view with the intellectual gaps alluded to

Yet a full course may not be necessary: For the past several years, the Intermediate Dynamics course at Notre Dame has started with an only 2-3 week presentation of linear algebra, both as a prelude to the three-dimensional dynamics to follow, and for its intrinsic pedagogical merit—to organize the bits and pieces of concepts into some organic whole However successful the latter goal has been, the former has proven beneficial

With regard to the other topic of Lagrangian mechanics, the situation is perhaps even more critical At a time when the analysis of large-scale systems has become increasingly important, the presentation of energy-based dynam-ical techniques has been surprisingly absent from most undergraduate texts altogether These approaches are founded on virtual work (not typically the undergraduate's favorite topic!) and not only eliminate the need to consider the forces at interconnecting pins [assumed frictionless], but also free the designer from the relatively small number of vector coordinate systems available to de-scribe a problem: he can select a set of coordinate ideally suited to the one at hand

With all this in mind, the following text commits to paper a course which

has gradually developed at Notre Dame as its "Intermediate Dynamics"

offer-ing It starts with a relatively short, but rigorous, exposition of linear systems, culminating in the diagonalization (where possible) of matrices—the foundation

of principal coordinates There is even an [optional] section dealing with Jordan normal form, rarely presented to students at this level In order to understand this process fully, it is necessary that the student be familiar with how the [ma-

trix] representation of a linear operator (or of a vector itself) changes with a

transformation of basis, as well as how the eigenvectors—in fact the new axes themselves—affect this particular choice of basis That, at least in the case

of real, symmetric, square inertia matrices, this corresponds to a rotation of

axes requires knowledge of axis rotation and the matrices which generate such rotations This, in turn, demands an appreciation of bases themselves and

XI

particularly, the idea of linear independence (which many students feel deals exclusively with the Wronskian) and partitioned matrix multiplication By the time this is done, little more effort is required to deal with vector spaces in general

This text in fact grew out of the need to dispatch a [perceived]

responsi-bility to rigor {Le.proofs of theorems) without bogging down class presentation

with such details Yet the overall approach to even the mathematical material

of linear algebra is a "minimalist" one: rather than a large number of arcane theorems and ideas, the theoretical underpinning of the subject is provided

by, and unified through, the basic theme of linear independence—the echelon

form for vectors and [subsequently] matrices, and the rank of the latter It can

be argued that these are the concepts the engineer and scientist can—should—

appreciate anyhow Partitioning establishes the connection between vectors and [the rows/columns of] matrices, and rank provides the criterion for the solution

of linear systems (which, in turn, fold back onto eigenvectors) In order to avoid the student's becoming fixated too early on square matrices, this fundamental theory is developed in the context of linear transformations between spaces of

arbitrary dimension It is only after this has been done that we specialize to square matrices, where the inverse, eigenvectors, and even properties of determi-

nants follow naturally Throughout, the distinction between vectors and tensors,

and their representations—one which is generally blurred in the student's mind

because it is so rarely stressed in presentation—is heavily emphasized

Theory, such as the conditions under which systems of linear equations have

a solution, is actually important in application But this linear algebra Part is more than mere theory: Linear independence, for example, leads to the con-cept of matrix rank, which then becomes a criterion for predicting the number

of solutions to a set of linear equations; when the cross product is shown to

be equivalent to a matrix product of rank 2, indeterminacy of angular

veloc-ity and acceleration from the rotating coordinate system equations in the next Part becomes a natural consequence Similarly, rotation matrices first appear

as an example of orthogonal matrices, which then are used in the tion of real symmetric matrices culminating the entire first Part; though this returns in the next Part in the guise of principal axes, its inverse—the rotation

diagonaliza-from principal axes to arbitrary ones—becomes a fundamental technique for the

determination of the inertia tensor

Given the audience for which this course is intended, the approach has been surprisingly successful: one still recalls the delight of one student who said that she had decided to attack a particular problem with rotation matrices, and "It worked!" Admittedly, such appreciation is often delayed until the part on rigid body dynamics has been covered—yet another reason for trying to have some

of the more technical detail in the text rather than being presented in class This next Part on 3-D dynamics starts with a relatively routine exposition of kinematics, though rather more detail than usual is given to constraints on the motion resulting from interconnections, and there is a perhaps unique demon-

stration that the fundamental relation dr = dO x r results from nothing more

than the fixed distance between points in a rigid body Theory from the first

Xll

Part becomes integrated into the presentation in a discussion of the

indetermi-nacy of angular velocity and acceleration without such constraints Kinetics is

preceded by a review of particle and system-of-particles kinetics; this is done

to stress the particle foundation on which even rigid body kinetics is based as

much as to make the text self-contained The derivation of the Euler tions is also relatively routine, but here the similarity to most existing texts

equa-ends: these equations are presented in matrix form, and principal coordinates

dispatched with reference to diagonalization covered in the previous Part The

flexibility afforded by this allows an arbitrary choice of coordinates in terms of

which to represent the relevant equations and, again, admits of a more

trans-parent comprehension than the norm 3-D rigid body machine kinetics, almost

universally ignored in elementary presentations, is also covered: the emphasis

is on integrating the kinematics and kinetics into a system of linear equations,

making the difference between single rigid bodies and "machines" quantitative

rather than qualitative There is also a rather careful discussion of the forces at

[ball-and-socket] connections in machines; this is a topic often misunderstood

in Statics, let alone Dynamics! While the gyroscope equations of motion are developed de nuvo as in most texts, they are also obtained by direct applica-

tion of the Euler equations; this is to overcome the stigma possibly resulting from independent derivation—the misconception that gyroscopes are somehow

"special," not covered by the allegedly more general theory There is a brief section detailing the use of the general equations of motion to obtain the vari-ational equations necessary to analyze stability Finally, a more or less routine treatment of energy and momentum methods is presented, though the imple-mentation of kinetic constants and the conditions under which vanishing force

or moment lead to such constants are presented and utilized; this is to set the scene for the next Part, where Analytical Dynamics employs special techniques

to uncover such "integrals of the motion."

That next Part treats Analytical Mechanics Lagrangian dynamics is oped first, based on Newton's equations of motion rather than functional mini-mization Though the latter is mentioned as an alternative approach, it seems

devel-a mdevel-ajor investment of effort—devel-and time—to develop one reldevel-atively devel-abstruse cept (however important intrinsically), just to derive another almost as bad by appeal to a "principle" (Least Action) whose origin is more teleological than physical! Rather more care than usual, however, is taken to relate the concepts

con-of kinematic "constraints" and the associated kinematical equations among the coordinates; this is done better to explain Lagrange multipliers Also included

is a section on the use of non-holonomic constraints, in good part to introduce Lagrange multipliers as a means of dealing with them; while the latter is typ-

ically the only methodology presented in connection with this topic, here it is

actually preceded by a discussion of purely algebraic elimination of redundant constraints, in the hopes of demonstrating the fundamental issue itself Withal,

the "spin" put on this section emphasizes the freedom to select arbitrary

coor-dinates instead of being "shackled" by the standard vector coordinate systems developed to deal with Newton's laws The discussion of the kinetic constants

of energy and momentum in the previous Part is complemented by a discussion

X l l l

of "integrals of the motion" in this one; this is the springboard to introduce cobi's integral, and ignorable coordinates become a means of uncovering other such constants of the motion in Lagrangian systems—at least, if one can find the Right Coordinates! The Part concludes with a chapter on Hamiltonian dynamics Though this topic is almost universally ignored by engineers, it is

Ja-as universal in application Ja-as Conservation of Energy and is the lingua franca

of Dynamical Systems, with which every modern-day practitioner must have some familiarity Unlike most introductions to the field, which stop after hav-ing obtained the equations of motion, this presentation includes a discussion of canonical transformations, separability, and the Hamilton-Jacobi Equation: the

fact that there is a systematic means of obtaining those variables—"ignorable" coordinates and momenta—in which the Hamiltonian system is completely sol- uble is, after all, the raison d^etre for invoking a system in this form, as opposed

to the previous Lagrangian formulation, in the first place Somewhat more attention to time-dependent Hamiltonian systems is given than usual Addi-tionally, like the Lagrangian presentation, there is a discussion of casting time

as an ordinary variable; though this is often touched upon by more advanced texts on analytical mechanics, the matter seems to be dropped almost immedi-ately, without benefit of examples to demonstrate exactly what this perspective entails; one, demonstrating whatever benefits it might enjoy, is included in this chapter

A N o t e on Notation We shall occasionally have the need to

distin-guish between "unordered" and "ordered" sets Unordered sets will be denoted

by braces: {a, 6} = {&, a}, while ordered sets will be denoted by parentheses: (a, b) ^ (6, a) This convention is totally consistent with the shorthand notation for matrices: A = (cnj)'

Throughout, vectors are distinguished by boldface: V; unit vectors are tinguished with a "hat" and are generally lower-case letters: i Tensor/matrix

dis-quantities are written in a boldface sans-serif font, typically in upper-case: I

(though the matrix representation of a vector otherwise denoted with a

lower-case boldface will retain the lower-case: v ~ t;)

Material not considered essential to the later material is presented in a smaller type face; this is not meant to denigrate the material so much as to provide a visual map to the overall picture Students interested more in ap-plications are typically impatient with a mathematical Theorem/Proof format, yet it surely is necessary to feel that a field has been developed logically For this reason, each section concludes with a brief guide of what results are used merely to demonstrate later, more important results, and what are important intrinsically; it is hoped this will provide some "topography" of the material presented

A N o t e on Style There have been universally two comments by students

regarding this text: "It's all there," and "We can hear you talking." In

retro-spect, this is less a "text" than lectures on the various topics Examples are

unusually—perhaps overwhelmingly!—complete, with all steps motivated and

XIV

virtually all intermediate calculations presented; though this breaks the page format" currently in favor, it counters student frustration with "terseness." And the style is more narrative than expository It is hoped that lecturers do not find this jarring

"one-R.A Rowland South Bend, Indiana

1.1.2 Subspaces of a Vector Space 12

1.2 The Basis of a Vector Space 13

1.2.1 Spanning Sets 14

1.2.2 Linear Independence 16

A Test for Linear Independence of n-tuples: Reduction to

Echelon Form 18 1.2.3 Bases and the Dimension of a Vector Space 28

Theorems on Dimension 28 1.3 The Representation of Vectors 32

1.3.1 n-tuple Representations of Vectors 34

1.3.2 Representations and Units 37

1.3.3 Isomorphisms among Vector Spaces of the Same Dimension 38

2 Linear Transformations on Vector Spaces 41

2.1 Matrices 43

2.1.1 The "Partitioning" and Rank of Matrices 44

The Rank of a Matrix 44 2.1.2 Operations on Matrices 48

Inner Product 49 Transpose of a Matrix Product 51

Block Multiplication of Partitioned Matrices 52 Elementary Operations through Matrix Products 54 2.2 Linear Transformations 61

xvi CONTENTS

Domain and Range of a [Linear] Transformation and their

Dimension 64 2.2.1 Linear Transformations: Basis and Representation 65

Dyadics 68 2.2.2 Null Space of a Linear Transformation 72

Dimension of the Null Space 74 Relation between Dimensions of Domain, Range, and Null

Space 74 2.3 Solution of Linear Systems 77

"Skips" and the Null Space 82 2.3.1 Theory of Linear Equations 84

Homogeneous Linear Equations 84 Non-homogeneous Linear Equations 85

2.3.2 Solution of Linear Systems—Gaussian Elimination 88

2.4 Linear Operators—Differential Equations 90

3 Special Case—Square Matrices 97

The "Algebra" of Square Matrices 98 3.1 The Inverse of a Square Matrix 99

Properties of the Inverse 102 3.2 The Determinant of a Square Matrix 103

Properties of the Determinant 105 3.3 Classification of Square Matrices I l l

3.3.1 Orthogonal Matrices—Rotations 113

3.3.2 The Orientation of Non-orthonormal Bases 115

3.4 Linear Systems: n Equations in n Unknowns 116

3.5 Eigenvalues and Eigenvectors of a Square Matrix 118

3.5.1 Linear Independence of Eigenvectors 123

3.5.2 The Cayley-Hamilton Theorem 128

3.5.3 Generalized Eigenvectors 131

3.5.4 Application of Eigenvalues/Eigenvectors 137

3.6 Application—Basis Transformations 140

3.6.1 General Basis Transformations 141

Successive Basis Transformations—Composition 144 3.6.2 Basis Rotations 151

3.7 Normal Forms of Square Matrices 156

3.7.1 Linearly Independent Eigenvectors—Diagonalization 157

Diagonalization of Real Symmetric Matrices 162 3.7.2 Linearly i^ependent Eigenvectors—Jordan Normal

Form 163

Epilogue 171

CONTENTS xvii

II 3-D Rigid B o d y Dynamics 173

Prologue 175

4 Kinematics 177

4.1 Motion of a Rigid Body 177

4.1.1 General Motion of a Rigid Body 178

Differentials 178 4.1.2 Rotation of a Rigid Body 180

Differential Rigid Body Rotation 181 Angular Velocity and Acceleration 185 Time Derivative of a Unit Vector with respect to Rotation 187

4.2 Euler Angles 191

4.2.1 Direction Angles and Cosines 192

Vector Description 192 Coordinate System Description 193

4.2.2 Euler Angles 194

Vector Description 194 Coordinate System Description 196

4.3 Moving Coordinate Systems 197

4.3.1 Relative Motion: Points 198

4.3.2 Relative Motion: Coordinate Systems 199

Time Derivatives in Rotating Coordinate Systems 200 Applications of Theorem 4.3.1 201 Rotating Coordinate System Equations 202

Distinction between the ">1/^" and "re/" Quantities 203 The Need for Rotating Coordinate Systems 209 4.4 Machine Kinematics 210

4.4.1 Motion of a Single Body 211

A Useful Trick 218 The Non-slip Condition 220

The Instantaneous Center of Zero Velocity 225 4.4.2 Kinematic Constraints Imposed by Linkages 228

Clevis Connections 228 Ball-and-socket Connections 237

4.4.3 MotionofM^tep/e Rigid Bodies ("Machines") 238

A Caveat regarding Conservation 262

xviii CONTENTS

5.1.2 Particle System Kinetics 263

Kinetics relative to a Fixed System 264

Kinetics relative to the Center of Mass 267 5.2 Equations of Motion for Rigid Bodies 272

5.2.1 Angular Momentum of a Rigid Body—the Inertia Tensor 273

Properties of the Inertia Tensor 280 Principal Axes 287 5.2.2 Equations of Motion 304

Forces/Moments at Interconnections 312 Determination of the Motion of a System 325 5.2.3 A Special Case—the Gyroscope 327

Gyroscope Coordinate Axes and Angular Velocities 327 Equations of Motion 328 Special Case—Moment-free Gyroscopic Motion 333

General Case—Gyroscope with Moment 338

5.3 Dynamic Stability 346

5.4 Alternatives to Direct Integration 352

5.4.1 Energy 353

Kinetic Energy 353 Work 355 Energy Principles 357

6 Analytical Dynamics: Perspective 389

6.1 Vector Formulations and Constraints 389

6.2 Scalar Formulations and Constraints 391

6.3 Concepts from Virtual Work in Statics 392

7 Lagrangian Dynamics: Kinematics 397

7.1 Background: Position and Constraints 397

Categorization of Differential Constraints 403 Constraints and Linear Independence 405 7.2 Virtual Displacements 408

7.3 Kinematic vs Kinetic Constraints 412

7.4 Generalized Coordinates 416

Derivatives of r and v with respect to Generalized

Coor-dinates and Velocities 420

CONTENTS xix

8 Lagrangian D y n a m i c s : Kinetics 423

8.1 Arbitrary Forces: Euler-Lagrange Equations 424

Notes on the Euler-Lagrange Equations 427 8.2 Conservative Forces: Lagrange Equations 443

Properties of the Lagrangian 453 8.3 Differential Constraints 455

8.3.1 Algebraic Approach to Differential Constraints 456

8.3.2 Lagrange Multipliers 458

Interpretation of the Lagrange Multipliers 465

8.4 Time as a Coordinate 467

9 Integrals of Motion 471

9.1 Integrals of the Motion 471

9.2 Jacobi's Integral—an Energy-like Integral 473

9.3 "Ignorable Coordinates" and Integrals 478

10.2.2 q and p as Lagrangian Variables 490

10.2.3 An Important Property of the Hamiltonian 492

10.3 Integrals of the Motion 495

Part I

Linear Algebra

Prologue

The primary motivation for this part is to lay the foundation for the next one, deahng with 3-D rigid body dynamics It will be seen there that the "iner-

tia," I, a quantity which is a simple scalar in planar problems, blossoms into a

"tensor" in three-dimensional ones But in the same way that vectors can be

represented in terms of "basis" vectors i, j , and fc, tensors can be represented as

3 x 3 matrices, and formulating the various kinetic quantities in terms of these matrices makes the fundamental equations of 3-D dynamics far more transpar-ent and comprehensible than, for example, simply writing out the components

of the equations of motion (Anyone who has seen older publications in which the separate components of moments and/or angular momentum written out doggedly, again and again, will appreciate the visual economy and conceptual clarity of simply writing out the cross product!) More importantly, however, the more general notation allows a freedom of choice of coordinate system While it is obvious that the representation of vectors will change when a dif-ferent basis is used, it is not clear that the same holds for matrix representations But it turns out that there is always a choice of basis in which the inertia tensor

can be represented by a particularly simple matrix—one which is diagonal in

form Such a choice of basis—tantamount to a choice of coordinates axes—is referred to in that context as "principle axes." These happen to be precisely the same "principle axes" the student my have encountered in a course in Solid Mechanics; both, in turn, rely on techniques in the mathematical field of "linear algebra" aimed at generating a standard, "canonical" form for a given matrix Thus the following three chapters comprising this Part culminate in a dis-cussion of such canonical forms But to understand this it is also necessary

to see how a change of basis affects the representation of a matrix (or even a vector, for that matter) And, since it turns out that the new principle axes are nothing more than the eigenvectors of the inertia matrix—and the inertia matrix in these axes nothing more than that with the eigenvalues arrayed along the diagonal—it is necessary to understand what these are (If one were to walk

up to you on the street as ask the question "Yes, but just what is an

eigenvec-tor?", could you answer [at least after recovering from the initial shock]?) To find these, in turn, it is necessary to solve a "homogeneous" system of linear equations; under what conditions can a solution to such a system—or any other linear system, for that matter—be found? The answer lies in the number of

"linearly independent" equations available And if this diagonalization depends

on the "basis" chosen to represent it, just what is all this basis business about anyhow?

In fact, at the end of the day, just what is a vector?

The ideas of basis and linear independence are fundamental and will pervade

all of the following Part But they are concepts from the field of vector spaces^

so these will be considered first That "vectors" are more general than merely the directed line segments—"arrows"—that most scientists and engineers are famihar with will be pointed out; indeed, a broad class of objects satisfy exactly

the same properties that position, velocity, forces, et al (and the way we add and

multiply them) do While starting from this point is probably only reasonable for mathematicians, students at this level from other disciplines have developed the mathematical sophistication to appreciate this fact, if only in retrospect

Thus the initial chapter adopts the perspective viewing vectors as those objects (and the operations on them) satisfying certain properties It is hoped that this

approach will allow engineers and scientists to be conversant with their more

"mathematical" colleagues who, in the end, do themselves deal with the same problems they do!

The second chapter then discusses these objects called tensors—"linear

trans-formations^^ between vector spaces—and how at least the "second-order" tensors

we are primarily interested in can be represented as matrices of arbitrary square) dimension The latter occupy the bulk of this chapter, with a discussion

(non-of how they depend on the very same basis the vectors do, and how a change

of this basis will change the representations of both the vectors and the tensors

operating on them If such transformations "map" one vector to another, the inverse operation finds where a given vector has been transformed from; this is precisely the problem of determining the solution of a [linear] equation

Though the above has been done in the context of transformations between

arbitrary vector spaces, we then specialize to mappings between spaces of the same dimension—represented as square matrices Eigenvectors and eigenvalues,

inverse, and determinants are treated—more for sake of completeness than thing else, since most students will likely already have encountered them at this stage The criterion to predict, for a given matrix, how many linearly indepen-dent eigenvectors there are is developed Finally all this will be brought together

any-in a presentation of normal forms—diagonalized matrices when we have able a full set of linearly independent eigenvectors, and a block-diagonal form when we don't These are simply the [matrix] representations of linear trans-formations in a distinguishing basis, and actually what the entire Part has been aiming at: the "principle axes" used so routinely in the following Part on 3-

avail-D rigid-body dynamics are nothing more than those which make the "inertial tensor" assume a diagonalized form

engineers, vectors are directed line segments—"arrows," as it were The same

motivation then makes more palatable a curious "arithmetic of arrows": we can find the "sum" of two arrows by placing them together head-to-tail, and

we can "multiply them by numbers" through a scaling, and possibly reversal

of direction, of the arrow It is possible to show from nothing more than this that these two operations have certain properties, enumerated in Table 1.1 (see page 7)

At this stage one typically sees the introduction of a set of mutually

perpen-dicular "unit vectors" (vectors having a magnitude of 1), i, j , and k—themselves

directed line segments One then represents each vector A in terms of its [scalar]

"components" along each of these unit vectors: A = A^i + Ayj -\- Azk (One might choose to "order" these vectors i, j , and k; then the same vector can be written in the form of an "ordered triple" A = (Ax.Ay^Az).) Then the very

same operations of "vector addition" and "scalar multiplication" defined for directed line segments can be reformulated and expressed in the Cartesian (or ordered triple) form Presently other coordinate systems may be introduced—in polar coordinates or path-dependent tangential-normal coordinates for example, one might use unit vectors which move with the point of interest, but all quan-tities are still referred to a set of fundamental vectors

[The observant reader will note that nothing has been said above regarding the "dot product" or "cross product" typically also defined for vectors There

is a reason for this omission; suffice it to point out that these operations are

defined in terms of the magnitudes and (relative) directions of the two vectors,

so hold only for directed line segments.]

6 CHAPTER 1 VECTOR SPACES

The unit vectors introduced are an example of a '^basis^^] the expression of

A in terms of its components is referred to as its representation in terms of that

basis Although the facts are obvious, it is rarely noted that a) the representation

of A will change for a different choice oft, j , and fc, but the vector A remains the

same—it has the same magnitude and direction regardless of the representation;

and b) that, in effect, we are representing the directed line segment A in terms

of real numbers Ax, Ay, and Az The latter is particularly important because it

allows us to deal with even three-dimensional vectors in terms of real numbers

instead of having to utilize trigonometry {spherical trigonometry, at that!) to

find, say, the sum of two vectors

In the third chapter, we shall see how to predict the representation of a vector given in one basis when we change to another one utilizing matrix products While talking about matrices, however, we shall also be interested in other applications, such as how they can be used to solve a system of linear equations

A fundamental idea which will run through these chapters—which pervades all

of the field dealing with such operations, "linear algebra"—is that of "linear

independence." But this concept, like that of basis, is one characterizing vectors

Thus the goal if this chapter is to introduce these ideas for later application Along the way, we shall see precisely what characterizes a "basis"—what is required of a set of vectors to qualify as a basis It will be seen to be far more general than merely a set of mutually perpendicular unit vectors; indeed, the entire concept of "vector" will be seen to be far broader than simply the directed line segments formulation the reader is likely familiar with

To do this we are going to turn the first presentation of vectors on its ear: rather than defining the vectors and operations and obtaining the properties in

Table 1.1 these operations satisfy, we shall instead say that any set of objects,

on which there are two operations satisfying the properties in Table 1.1, are vectors Objects never thought of as being "vectors" will suddenly emerge

as, in fact, being vectors, and many of the properties of such objects, rather than being properties only of these objects, will become the common to all

objects (and operations) we define as a ''vector space.^^ It is a new perspective,

but one which, with past experience, it is possible to appreciate in retrospect Along the way we shall allude to some of the concepts of "algebra" from the field of mathematics Though not the sort of thing which necessarily leads

to the solution of problems, it does provide the non-mathematician with some background and jargon used by "applied mathematicians" in their description

of problems engineers and scientists actually do deal with, particularly in the

field of non-linear dynamics—an area at which the interests of both intersect

30 G 7^ : t; 0 0 = i; for all i; G 7^ (additive identity element)

for each v e V^3v e V \v ^v = ^ (additive inverse)

Table 1.1: Properties of Vector Space O p e r a t i o n s

1.1.1 T h e "Algebra" of Vector Spaces^

There are several things to be noted about this table The first is that the five "G" properties deal only with the " 0 " operation The latter "V" properties relate that operation to the " 0 " one

"Closure" is generally regarded by non-mathematicians as so "obvious" that it

hardly seems worth mentioning Yet, depending on how the set ^ is defined,

reason-ably non-pathological cases in which this is violated can be found But in properties

(V2) for example, (vi0V2) is an element of ^ by (Gl), while (a 0 v) and (h 0 v) are

by (VI); thus we know that the " 0 " operation on the left-hand side of this equation

is defined (and the result is known to be contained in 7^), while the two terms on the right-hand side not only can be added with 0 , but their "vector sum" is also in ^ It seems a little fussy, but all legal (and perhaps somewhat legalistic)

In the same way, to make a big thing about "associative" laws always seems a little strange But remember that the operations we are dealing with here are "binary": they

operate on only two elements at a time The associative laws group elements in pairs

(so the operations are defined in the first place), and then show that it makes no difference how the grouping occurs

It is important to note that there are actually four types of operations used in this

table: in addition to the 0 operation between numbers and vectors, and the 0 between vectors themselves, there appear "+" and "•" between the real numbers; again, we are just being careful about which what operations apply to what elements Actually, what all of the equalities in the Table imply is not just that the operations on each

side are defined, but that they are, in fact, equall

^ Note (see page xiii in the Foreword) that "optional" material is set off by using a smaller

typeface

8 CHAPTER 1 VECTOR SPACES

Property (V5) appears a little strange at first What is significant is that the

number " 1 " is that real number which, when it [real-]multiplies any other real number,

gives the second number back—the "multiplicative identity element" for real numbers

That property states that " 1 " will also give v back when it scalar multiplies v

It all seems a little fussy to the applied scientist and engineer, but what is really

being done is to "abstract" the structure of a given set of numbers, to see the essence

of what makes these numbers combine the way they do This is really what the term

"algebra" (and thus "linear algebra") means to the mathematician While it doesn't actually help to solve any problems, it does free one's perspective from the prejudice of experience with one or another type of arithmetic This, in turn, allows one to analyze the essence of a given system, and perhaps discover unexpected results unencumbered

by that prejudice

In fact, the "G" in the identifying letters actually refers to the word "group"—

itself a one-operation algebraic structure defined by these properties The simplest

example is the integers: they form a group under addition (though not multiplication:

there is no multiplicative inverse; fractions—rational numbers—are required to allow

this) In this regard, the last property, (G5), defined the group as a "commutative" or

"Abelian" group We will shortly be dealing with the classic instance of an operation

which forms a group with appropriate elements but which is not Abelian: the set of

n X n matrices under multiplication The [matrix] product of two matrices A and B

is another square matrix [of the same dimension, so closed], but generally AB y^ BA

One last note is in order: All of the above has been couched in terms of "real numbers" because this is the application in which we are primarily interested But,

in keeping with the perspective of this section, it should be noted that the actual definition of a vector space is not limited to this set of elements to define scalar multiplication Rather, the elements multiplying vectors are supposed to come from a

"field"—yet another algebraic structure Unfortunately, a field is defined in terms of

still another algebraic structure, the ^^ring^^:

Recall that a vector space as defined above really dealt with two sets of elements: elements of the set 1^ itself, and the [set of] real numbers; in addition, there were two operations on each of these sets: "addition," "©" and "+" respectively, and

"multiplication," "•" on the reals and " 0 " between the reals and elements of 1/ In

view of the fact that we are examining the elements which "scalar multiply" the vectors,

it makes sense that we focus on the two operations "+" and "•" on those elements:

Definition 1.1.1 A ring is a set of elements ^ and two operations "+" and "•" such

that elements of i ^ form an Abelian group under "+," and, for arbitrary vi, 7*2, and

7*3 in ^ , "•" satisfies the three properties in Table 1.2:

(RMl) ri-r2 e ^ (closure)

(RM2) (ri ' r2) ' rs = ri ' {r2 ' rs) (associativity)

(RM3) (ri + r2) • ra = (ri • rs) + (r2 • rs) (distribution)

(RM4) r i • (r2 + rs) == (ri • r2) + (ri • rs) (distribution)

Table 1.2: Properties of Multiplication in Rings

Note that the "•" operation is not commutative (though the "+" operation is, by

the ring definition that the elements and "+" form an Abelian group); that's why

1.1 VECTORS 9

there are two "distributive" laws Again we return to the example of square matrices:

we can define the "+" operation to be an element-wise sum, and the "•" operation

to be ordinary matrix multiplication; then matrices, under these two operations, do

in fact form a ring (in which, as noted above, the multiplication operation is

non-commutative)

Now that we have dispatched the ring, we can finally define the field:

Definition 1.1.2 A field is a ring whose non-0 elements form an Abelian group under

multiplication ("•")

Recall that "0" is the additive identity element—that which, when "added" to any

element returns that same element; thus, in eff'ect, this definition is meant to ensure

that the "reciprocal" of an element exists while precluding division by 0 Clearly, real

numbers (under the usual definitions of addition and multiplication) form a field; so do

the rational numbers (fractions) On the other hand, the set of integers (positive and

negative ones—otherwise there is no "additive inverse" for each) form a ring under the

usual operations, but not a field: there is no multiplicative inverse without fractions

Having finally sorted out what a "field" is, we are ready to view vector spaces

from the most general perspective: The elements used to scalar multiply element of

the vector space must arise from a field, ^ In this context, we refer to the vector

space as consisting of elements from one set, 7^, on which vector addition is defined,

in combination with another set, ^ , from which the "scalars" come in order to define

scalar multiplication; the resulting vector space is referred to as a vector space over the

field 7 when we wish to emphasize the identity of those elements which scalar multiply

the "vectors." In all the below examples, and almost exclusively in this book, we shall

be talking about vector spaces over the reals Again, while a vector space consists

technically of the set V of "vectors," the field J of "scalars" [and the operations

of "addition" and "multiplication" defined on it\\ and the two operations, " 0 " and

" 0 " of "vector addition" on V and "scalar multiplication" between jF and V—which

really comprises another set of two spaces and two operations (1^, jT, 0 , 0 ) — w e shall

typically employ the more economical [and perhaps abusive] description of simply "the

vector space ^ " where the field and operations are all implicitly understood This is a

reasonably common practice and exactly what was done in the examples below One

must keep in mind, however, that there are really four types of elements required to

define a vector space completely, and that the various operations on the sets V and

jT are distinct

Now having this general definition of a vector space, we are a r m e d t o see how

m a n y familiar m a t h e m a t i c a l quantities actually qualify as such a space Recall

t h a t in order t o define t h e space in question, we are obliged t o define 1) t h e

elements a n d 2) t h e [two] operations; if t h e y satisfy t h e properties in Table 1.1,

t h e y are, in fact, a vector space T h o u g h these examples will be identified by

t h e elements comprising t h e space, t h e actual vector space involves b o t h t h e

elements and t h e operations on it

E x a m p l e 1 1 1 Directed Line Segments: under t h e operations of head-to-tail

"addition" and "scalar multiplication" by t h e reals This is t h e classic case,

characterized by position, velocity, force, etc P

^To set off the Examples, they are terminated by the symbol "|.'

10 CHAPTER! VECTOR SPACES

Example 1.1.2 Expressions of the form aiUi + a2U2 + + anUn'- [Note

that we are not calling the Ui vectors^] If we define "addition" of such forms

(aiUi + a2U2 + + anUn) 0 (hiui + h2U2 + + hnUn) =

(ai + bi)ui + (a2 + b2)u2 H \- {an + bn)un,

and "scalar multiplication"

c 0 {aiUi + a2U2 + + anUn) = (c • ai)ui + (c • a2)u2 H h (c • an)tXn,

the properties defining a vector space are preserved (Note that closure results

from the definitions sending one form into the same form.) \

The Ui here act merely as "place keepers." This is effectively the same role

played by the ordered nature of the next example:

Example 1.1.3 [Ordered] Row/Column n-tuples: Our "vectors" consist of

arrays (here columns) of real numbers We define (almost obviously) addition

Example 1.1.4 m x n Matrices: Simply generalizing the above definitions

for addition—note that these arrays are "ordered" just like their row or column

\bml bm2 ' ' ( a i i + & i i

—we see that such objects, too, form a vector space |

The above examples are almost mundane But now we shall see several

more examples of "function spaces" showing that even certain functions can be

viewed as being vector spaces

Example 1.1.5 n^^-degree Polynomials: We regard polynomials of [at most]

n^'^-degree to be the elements:

f{x) = ao-\- aix + a2X^ + + anX^

and define, again obviously, our addition:

fi{x) e /2(x) = (ao + 6o) + (ai + hi)x + (a2 + h2)x^ + + (an + 6n)x^

and scalar multiplication

c 0 f{x) = c • ao + c • aix + c • a2X^ + + c • anx'^

Again, we have a vector space (Note that both operations give n^'^-order

polynomials—i.e they are closed.)

Once again, the x^ effectively play the role of mere place keepers |

Example 1.1.6 Solutions to Homogeneous n^^-order Linear Differential

Equa-tions: Given the n*^-order linear ordinary differential equation in independent

variable t

xM + an-i(t)xt^-il + • + ao{t)x - 0,

because of the linearity., any two solutions xi{t) and X2(^), under the above

def-initions of [functional] "vector addition" and "scalar multiplication" will satisfy

this equation Thus these operations satisfy closure and we can regard the set of

all such solutions [to this system] as comprising a vector space This important

topic will return in Section 2.4 |

Example 1.1.7 Periodic Functions with Period T: Without belaboring the

point any further, such functions are closed under the type of functional addition

and multiplication by scalars defined above—they are still periodic with period

T—and thus form a vector space |

The last example will comprise the entire next chapter:

Example 1.1.8 Linear Transformations from One Vector Space to Another:

In order to discuss this, we must first define what we mean by a "linear

trans-formation" :

12 CHAPTER! VECTOR SPACES

Definition 1.1.3 A linear transformation from a vector space l/i to another

vector space ^2 is a function, L, on l/i taking Vi and t;2 G 'I^i:

L{avi + bv2) = aL{vi) + 6L(i;2) (1-1-1)

(Note that the the Vi and "+" on the left-hand side of this equation refer to

l/i] in particular, the "+" is "vector addition" on ^i On the right-hand side,

on the other hand, L{vi) are in ^2 and "+" is "vector addition" defined in

that space Similarly, the "scalar products" one each side are those appropriate

to the two spaces [To be technical in the spirit of the previous discussion of

"fields," we must have both vector spaces defined over the same field!])

For any such linear transformations L, Li and L2, we can define vector

addition of two such transformations

{Li Q L2){v) = Li{v) + L2{v)

and scalar multiplication

( c 0 L ) = cL{v)

(Again, the "addition" and "scalar multiplication" on the right-hand side of

the above two defining equations are those in ^ ) By the definition of linear

transformations (1.1-1), (Li 0 L2) and (c 0 L) will also satisfy this definition

and be linear transformations; thus the above definitions of vector addition and

scalar multiplication are closed, and satisfy the defining properties of a vector

space I

Homework:

1 Show the set of linear transformations comprises a vector space

Many of the above examples might appear at first to be somewhat artificial

The point is, however, that they all, at least formally, satisfy the criteria of

being a vector space, and we shall return to them later to exemplify the concept

of basis

1.1.2 Subspaces of a Vector Space

We have seen that a vector space consists of several components: a set of

el-ements comprising the space itself, a "field" of elel-ements (along with its own

operations) with which we can "scalar multiply" elements of the vector space,

and a pair of operations defining "vector addition" and "scalar multiplication"

which satisfy certain criteria Typically, we concentrate on the first,

assum-ing the others implicitly We shall briefly focus on the identity of that set of

elements here, too

Given the field and operations, all elements of a vector space satisfy the

properties of Table 1.1; in particular, application of the operations to elements

1.2, THE BASIS OF A VECTOR SPACE 13

of the vector space result in other elements in the vector space—the closure

properties (Gl) and (VI) in the Table But it might happen that a certain

subset of elements in the space, identified by some criterion to be a "proper

subset" of that space (so forming a "smaller" set not equal to the original

space) will, when operated upon by the same operations, give back elements in

the subset A brief example might be in order to illustrate this idea:

Example 1.1.9 Consider the space of row 4-tuples—elements of the form

(t'l, f2, f3, V4)—under the operations defined for such elements in Example 1.1.3

Elements of the form (fi,f2,0,'^4) are clearly elements of this space; but the

results of vector addition and scalar multiplication on such elements return an

element of exactly the same form, \

Clearly such elements satisfy properties (G2)-(G5) and (V2)-(V5) in the

Table: they inherit these properties by virtue of being elements of the vector

space in the large But they also enjoy a unique closure property in that they

remain in the subset under the general operations Such elements then, through

this closure property, form a vector space in their own right: they satisfy all the

properties of the Table including closure Elements which retain their identity

under the general operations seem special enough to warrant their own category:

Definition 1.1.4 A [proper] subset of elements of a vector space which remains

closed under the operations of that vector space are said to form a [vector]

subspace of that space

Though at this stage such elements and the "subspace" they comprise might

seem merely a curiosity, they are relatively important, and it is useful to have

a criterion to identify them when they occur Since it is the closure property

which characterizes them, it is no surprise that the test depends only on closure:

Theorem 1.1.1 A subset of a vector space forms a subspace if and only if

{{a ^ Vi) Q {b(^ V2)), for all "scalars" a and b and all vi and V2 in the subspace,

is itself an element of the subspace

Homework:

1 Prove this

1.2 The Basis of a Vector Space

This chapter has been motivated by appeal to our experience with "Cartesian"

vectors, in which we introduce the fundamental set of vectors i, j , and k in terms

of which to represent all other vectors in the form A = A^l + Ay^ + A^h [Note

that after all the care exercised in the previous section to differentiate between

the scalar "+" and "•" and vector " 0 " and " 0 , " we are reverting to a more

14 CHAPTER 1, VECTOR SPACES

common, albeit sloppier, notation The distinction between these should be clear from context.] In terms of the "algebra" discussed in the previous section,

we see that each part of this sum is the scalar products of a real with a vector (so

a vector by Property (VI) in Table 1.1), and the resulting vectors are summed

using vector addition, so yielding a vector (by Property (Gl) of the same Table);

this is the reason the two defining operations on vectors are so important Such

a vector sum of scalar products is referred to as a linear combination of the

vectors

While these fundamental vectors i, j , and k form an "orthonormal set"— mutually perpendicular unit vectors—we shall examine in this section the ques-

tion of how much we might be able to relax this convention: Do the vectors

have to be unit vectors? Or perpendicular? And how many such vectors are

required in the first place? All of these questions revolve around the issue of a

basis for a vector space

In essence, a basis is a minimal set of vectors, from the original vector space, which can be used to represent any other vector from that space as a linear

combination Rather like Goldilocks, we must have just the right number of

vectors, neither too few (in order to express any vector), nor too many (so that

it is a minimal set) The first of these criteria demands that the basis vectors (in reasonably descriptive mathematical jargon) span the set; the second that they be linearly independent The latter concept is going to return continually

in the next chapter and is likely one of the most fundamental in all of linear algebra

1.2.1 Spanning Sets

Though we have already introduced the term, just to keep things neat and tidy, let us formally define the concept of linear combination:

Definition 1.2.1 A linear combination of a set of vectors {t^i,'y25 • • • j'^n} is

a vector sum of scalar products of the vectors: aiVi -\- a2V2 + +

anVn-We note three special cases of this definition given the set {t;!, t;2, • • •,

^n}-1 Multiplication of a [single] vector by a scalar: Vi can be multiplied by a scalar c by simply forming the linear combination with ai = c and GJ = 0 for j ^ i

2 Addition of two vectors: Vi + Vj is just the linear combination in which

ai = a j = 1 and a/e = 0 for fe 7^ i , j

Thus far, there's nothing new here: the above two are just the fundamental operations on vector spaces, cast in the language of linear combinations But:

3 Interchange of two vectors in an ordered set: If the set is an ordered set of

vectors ( t ; i , , t;^, , 1;^, ) The claim is that we can put this into the

1.2 THE BASIS OF A VECTOR SPACE 15

form ( t ; i , , -Uj, , t;^, ) through a sequence of hnear combinations:

{Vi, ,Vi, ,Vj, ) ^ {Vi, ,Vi, ,Vj+Vi, )

- > {Vi, ,-Vj, ,Vi, )

- > {Vi, ,Vj, ,Vi, )

Thus the concept of "hnear combination" is more general that the definition

alone might suggest

With this, it is easy to formalize the concept of spanning:

Definition 1.2.2 A set of vectors {tii, 1*2, , n ^ } in a vector space 1/ is said

to span the space iff every v E ^ can be written as a linear combination of the

It's: V = ViUi + V2U2 + +

VnUn-We give two examples of this The first is drawn from the space of directed

line segments We purposely consider non-orthonormal vectors:

Example 1.2.1 Directed Line Segments: Consider, in the plane containing

them, the directed line segment vectors n i , U2 with

lengths 1 and 2 respectively and making an angle of

60° with each other, and a vector A of length 2 ing an angle of 135° with Ui We are asked to express

mak-A as a linear combination of Ui and U2

We desire Ai and A2 such that A = AiUi-\-A2U2'

Since we are dealing with directed line segments, it is necessary to use the "head-to-tail" definition of vec-tor addition, shown at right in the diagram above

Since we know all the angles in the resulting triangle, trigonometry tells us that

or, since \\AiUi\\ = \Ai\

sin 60° sin 75° sin 45°

Solving, we get that \Ai\ = 2.231 and 1^421 = 0.816 Taking into account that

AiUi reverses the direction of Ui so that Ai must be negative, we obtain finally

that A = -2.231ni -[- 0.816 U2 \

Note in the above example that the 0-vector—^just a point—can only be

expressed as the linear combination 0 Ui + 0 1^2: there is no way that the two

n's can be added together with non-0 coefficients to close the vector triangle

The next example considers column 2-tuples to be the vector space:

^Note the distinction between the norm—the magnitude—of a vector, ||.||, and the absolute

value, |.|, of a scalar

16 CHAPTER 1 VECTOR SPACES

Example 1.2.2 n-tuples (Ovev-spanned Space): Consider the vectors

1\ / 0 \ / I

"^"loj ^ ^ " l i j ^^"vi

and the vector in this space

.2

Express i; as a linear combination of the n's

Solution Again, we seek the scalars vi, V2, and vs such that

or, by the definitions of both scalar multiplication and vector addition for this

space

In this case—we have only two equation through which to find three unknowns—

there is no unique solution^; indeed, any vi and f 2 such that, for arbitrary value

of f3, satisfy vi — 3 — ^3 and V2 = 2 — vs will work! Thus, for example,

V = 2ui + 1 ^ 2 + ^ 3 = 4ui + 3^2 — ^3 = .etc

In this case, the 0-vector, simply the 2-tuple

can be expressed using any vi — —v^ and v^ — —v^ \

In both the examples above, the given vectors spanned the respective spaces;

in one case the trigonometry yielded a unique linear combination, while in the

other solution of the linear system did not The problem with the second case

is obvious: there were "too many" vectors; indeed, for this example u^ was

superfluous, in the sense that ^3 = 161+^2 But this is not an issue of simply

having "the same number of spanning vectors as the -dimension' of the space"

(whatever the latter means) since, for example, had one of the two vectors in

the first example been the 0, they wouldn't have spanned the space at all! Both

of these ideas are unified in the next section

1.2.2 Linear I n d e p e n d e n c e

In the second example above, we observed that 1x3 could be expressed in terms

of Ui and U2\ i.e., there were ci and C2 such that we could write

tX3 = CiUi +

C2U2-^See Section 2.3

1.2 THE BASIS OF A VECTOR SPACE 17

But why select U3, when we could just as easily have selected one of the others

to be solved for in terms of the remaining two? In order to avoid distinguishing

one vector from the rest, note that we could write the above equation in a

slightly more symmetric form:

ciui 4- C2U2 - Us = 0

—one which more equitably treats the three vectors, yet is true if and only if

one can be solved for in terms of the others This leads to the formal definition

of linear independence:

Definition 1.2.3 A set of vectors {'yi,'i^2, • • • ^'^n} is linearly dependent iff

there are {ci, C2, , Cn} not all zero such that ciVi + C2V2 + + CnVn = 0

Clearly, if all the Q in the above linear combination are 0, each term in

the vector sum will vanish and so will the sum itself; this is the reason for the

stricture that they not all vanish Again, we see that this is equivalent to being

able to solve for one vector in terms of the others: if, in particular, Q^ 7^ 0, the

above allows us to solve for Vi^ in terms of the other Vi:

Vi, = — y^Q^^^

A linearly independent set is, quite simply (and logically), one that is not

linearly (dependent! But, in application, it is useful to state the contrapositive

of the above definition explicitly:

Definition 1.2.4 A set of vectors {'yi,'y2, • • • ,'^n} is linearly independent iff

ciVi 4- C2V2 + + CnVn = 0 implies that Ci = 0 for all i

Now we see the importance of the observation in the Examples of the

pre-vious section that 0 could not, or could, respectively, be written as a linear

combination of the constituent vectors: this is precisely the test for linear

in-dependence! Using the above definition, it is easy to show the following useful

results:

T h e o r e m 1.2.1 Any set of vectors {i^i, 1^2, • • •, '^n} containing the zero vector

0 is linearly dependent

T h e o r e m 1.2.2 Any subset of a set {'yi, t ; 2 , , Vn} of linearly mdependent

vectors is also linearly independent

Homework:

1 Prove these two Theorems

18 CHAPTER 1 VECTOR SPACES

Warning: Many students may first have seen "linear dependence"

in a course on differential equations, in connection with the

"Wron-skian" in that field Reliance on that criterion alone as a

"quick-and-dirty" criterion for linear independence is shortsighted however: In

the first place, it evaluates the determinant of numbers, useful only

in the case of row or column n-tuples, Example 1.1.3 above; how, for example, could we have used this to show the linear independence of directed line segments Example 1.2.1? Secondly, even if our vector

space does consist of such n-tuples with numerical components, the determinant is only defined for a square matrix; what if one is ex-

amining the linear independence oi fewer (or more) than n vectors?

The above definitions, however, work in both cases

We shall now consider another criterion for linear independence Unlike the above definition, however, it will be applicable only to the n-tuple row or column vectors, and thus be apparently limited in scope Actually, as we shall see, it

can be applied to any vectors (Theorem 1.3.2); and in the discussion of matrices

[of real numbers], it is of fundamental utility

A Test for Linear Independence of n-tuples: Reduction to Echelon Form

The rationale for this test is to consider a set of n-tuple vectors and determine

the Ci in the definition, 1.2.4, through a systematic process rather than by solving

a set of linear equations as done in Example 1.2.2 This process, in turn, reduces

the set of vectors to a standard form, echelon form, whose individual vectors

reflect precisely the linear independence of the original set The technique rests

1.2 THE BASIS OF A VECTOR SPACE 19

The idea is to examine the hnear independence of the set of vectors by trying

to reduce it, through a sequence of hnear combinations of the vectors among

themselves, to the above "echelon^^^ form This can best be seen by an example:

Example 1.2.3 Reduction to Echelon Form: Investigate the linear

indepen-dence of the vectors

Solution Noting the "standard form" of the above vectors (1.2-1), we get a 1 in

the first entry in i;i and 0 in that entry in the rest by performing the following

operations—linear combinations of the vectors with vi:

v[ = Vi/3: ; V2 - 3t;i: ^ 3 - ' ^ i : V4 + 2i;i:

—then get 1 in the second entry of the second vector and corresponding O's in

the rest with V2:

-and finally, since ^3 = 0, eliminate the third entry in the others with ^4:

v'i' = < - < / 4 8 : v'i' = v'i + 3 < / 1 6 : < = v'^: < = < / 1 6 :

0 • V2 = {vs - vi) - {V2 - 3vi) = t;3 - t;2 + 2t;i (1.2-2)

^from the French echelon^ literally "rung of a ladder." Note that many use the term

^'reduced echelon" form to refer to that in which the leading term in normalized We invariably

normalize so do not make such a distinction

20 CHAPTER 1, VECTOR SPACES

Note that even if the order of the Vi had been different—say that Vs and V4

had been switched—the same [now fourth] vector would have vanished, giving

the correspondingly same linear combination Furthermore, the "echelon form"

at the end would be exactly the same except for the ordering | Now having exemplified the process, we must justify it The technique is

somewhat indirect, forming linear combinations of the vectors in the set to

yield either [linearly independent] vectors in the form of Theorem 1.2.3 or O's

Further, we are concluding that the vectors in the original set corresponding

to the new, linearly combined, vectors, are precisely those which are linearly

independent Clearly what is necessary first is a theorem like the following:

Theorem 1.2.4 A set of vectors t ; i , i ; 2 , ,t^n are linearly independent iff

there exist n linearly independent linear combinations Vi = J2j ^ij'^j ^^ ^^^

vectors

In fact, this is exactly what we shall prove But before doing so, we need

another fundamental result:

Theorem 1.2.5 In a space spanned by the n linearly independent vectors

Vi^ i = 1 , , n, there exist at most n linearly independent linear combinations

^i — X^?=i ^ij'^j ^^ them {Note: this does not say the Vi are unique—there

can be any number of different {Vi}\ only that each such set contains at most

n linearly independent vectors!)

Proof We know at least one set of n linear combinations—the {vi^ i — 1 , , n}

themselves—are linearly independent; assume [they or any other linear

combi-nations] {V^ = Yl^=i ^ij'^j^'^ — 1 • • •'^} are linearly independent and consider

the set comprising another V^+i = S ? = i ^(n+i)j'^i amended to them Is this

new set linearly dependent or independent; i.e., under what circumstances does

S r = i ^i^i vanish? Note that we can assume Vn+i 7^ 0; otherwise the amended

set would be linearly dependent already, by Theorem 1.2.1 In the same spirit,

we can also assume Cn+i 7^ 0 in this sum: if it did, the linear independence of

the original Vi would force their Ci to vanish Thus we can "normalize" the

linear independence criterion, Yl^=i ^i^i = 0, to J27=i ^i^i + ^ n + i = 0 by

dividing the first by Cn+i 7^ 0

Now if this last form vanishes,

We see immediately that the Ci cannot vanish for all i; if they did, the sum in

parentheses would vanish, collapsing the entire last sum to Vn+i and so forcing

Vn+i = 0 and violating our assumption that it isn't Thus there are c^ 7^ 0

such that the first sum in the series vanishes, making {V^, i = l , , n 4 - l }

1.2 THE BASIS OF A VECTOR SPACE 21

linearly dependent, and proving that the original n are the most which can be

linearly independent D

Note: This is a fundamental result—one which will return in our discussion

of "dimension" of a vector space in Section 1.2.3

We are now ready to prove the above theorem:

Proof of Theorem 1.2.4- The "implication" is trivial: Assume the {vi} are

lin-early independent; then the linear combinations

V i = (l)t;i + (0)t;2 + • • • + (0)i;n

V2 - (0)t;i + (l)t;2 + • • • + (0)t;n

Vn = {0)vi + (0)t;2 + • • • + {l)Vn

are linearly independent

Conversely, assume the set {Vi = Yl- VijVj^i = 1 n} are linearly

inde-pendent, and consider the two spaces

V = {all vectors spanned by the Vi^ i = 1 , , n}

^ = {all vectors spanned by the Vi, i = 1 , , n }

While all n of the Vi are linearly independent by hypothesis, at most n of the Vi

are; whatever number are is also the maximum number of linearly independent

linear combinations z^ can support, by the above Theorem But the vectors

{Vi, 2 = 1 , , n} C ^ ; and linear combinations of linear combinations of the

Vi are themselves linear combinations of the Vi, so 1/ C v Thus the nV i^ V.,

so at least n of the spanning Vi must be linearly independent, or Theorem 1.2.5

is violated Thus all n of the Vi must be linearly independent D

This shows that, in the Example 1.2.3 above, having obtained three linearly

independent linear combinations (the v'/') of the original vectors Vi, three of

the latter were, in fact, linearly independent But it doesn^t show that the first,

second, and fourth were the linearly independent ones This is a result of the

systematic way the echelon form was obtained:

Recall that we seek to have the i*^ vector have a 1 in its i*^ row, with O's

above and below But the O's above the i^^ row result from linear combinations

of vectors preceding the i^^ vector, while those below come from linear

combi-nations with those following it If a vector is "cleared" in the process of trying

to zero out entries above the i*^, such as the the third vector was in the second

iteration in Example 1.2.3 was, it is the result of hnear combinations with

vec-tors only preceding it] we have yet to operate on entries below the i*^ using the

following ones This is shown explicitly in (1.2-2) in the example Thus Vs is

linearly dependent on t?i and V2 but not on V4 [yet]

^Following common practice, proofs are set off by terminating them with a symbol, here

22 CHAPTERL VECTOR SPACES

But if we are not interested in which vectors are linearly independent (and typically, in application, we aren't), all that is necessary is to obtain the number

of linearly independent linear combinations, regardless of the order in which the original set was presented This means that we can "switch" the order of vectors

at will in order to standardize that order (Recall the remark on page 14: such

an interchange is itself a linear combination!) In particular, this "standardized order" in the present context is one in which the i^^ vector is reduced to a " 1 " in

its i*^ row, with O's below, precisely that in Theorem 1.2.3 This is the approach that will be followed in the sequel

Having exemplified the procedure and stated the theory on which it rests,

we can now organize it into a systematic technique for examining the linear

independence of a set of m vectors This will be done inductively, assuming that we have already reduced i vectors in the set to the form (1.2-1) through the i^^ row:

i+1 linearly independent

(in which the j ^ ^ component of Vi is vij)^ and are dealing with putting the

(i + 1)^* vector in standard form (Initially, i = 0.) If f^+i,^+i 7^ 0, we proceed

as usual By performing linear combinations of the preceding i vectors, this set

gets reduced to an echelon form

It might be the case that the (i +1)^* component of t;^+i, 1^^+1,^+1, is, in fact, 0;

in fact, it might be the case that fi+i,j = 0 for all j > i—i.e., that Vi^^i = 0—

as happened with v';^ in Example 1.2.3 above, in which case Vi^i is a linear combination of the Vj,j < i to the left of t^i+i, and we have shown that only i

1.2 THE BASIS OF A VECTOR SPACE 23

of the i + 1 vectors thus far put in standard form are Unearly independent In

any event, if we are only interested in how many hnearly independent vectors there were originally, we are free to reorder the set by switching Vi^i with one

of the Vj in which fj,i+i ^ 0 (Note that the two vectors being transposed have,

to this point, been operated on only by those to the left of t^i+i.) We then

normalize that entry and use it to eliminate the (i + 1)^* entries in the other vectors

But it is conceivable that, in fact, there is no other such vector with a non-0

{i + ly^ entry; that, at this stage in the procedure, the vectors have assumed

i + 1 linearly i n d e p e n d e n t using row i+2\

with a 0 in the (i +1)^* row of not only the (i +1)^* vector, but also in all others

to its right We briefly sketch an example of such a case:

Example 1.2.4 ^'Skip" case: Reduce the vectors

V 5 /

,VB

/ 1 \

3 -5

0

29

to [reduced] echelon form

Results of the obvious operations using, respectively,