Linear Algebra and Analytic Geometry for Physical Sciences

344 Pages·2018·2.82 MB·English

by  Giovanni Landi

Giovanni Landi Alessandro Zampini
(cid:129)
Linear Algebra and
Analytic Geometry
for Physical Sciences
123
GiovanniLandi Alessandro Zampini
University of Trieste INFNSezione diNapoli
Trieste Napoli
Italy Italy
ISSN 2192-4791 ISSN 2192-4805 (electronic)
Undergraduate Lecture Notesin Physics
ISBN978-3-319-78360-4 ISBN978-3-319-78361-1 (eBook)
https://doi.org/10.1007/978-3-319-78361-1
LibraryofCongressControlNumber:2018935878
©SpringerInternationalPublishingAG,partofSpringerNature2018
Contents
1 Vectors and Coordinate Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Applied Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 More Vector Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4 Divergence, Rotor, Gradient and Laplacian. . . . . . . . . . . . . . . . 15
2 Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1 Definition and Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 Vector Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.3 Linear Combinations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.4 Bases of a Vector Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.5 The Dimension of a Vector Space . . . . . . . . . . . . . . . . . . . . . . 33
3 Euclidean Vector Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.1 Scalar Product, Norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.3 Orthonormal Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.4 Hermitian Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.1 Basic Notions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.2 The Rank of a Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.3 Reduced Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.4 Reduction of Matrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.5 The Trace of a Matrix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5 The Determinant. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.1 A Multilinear Alternating Mapping . . . . . . . . . . . . . . . . . . . . . 69
5.2 Computing Determinants via a Reduction Procedure. . . . . . . . . 74
5.3 Invertible Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
6 Systems of Linear Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
6.1 Basic Notions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
6.2 The Space of Solutions for Reduced Systems. . . . . . . . . . . . . . 81
6.3 The Space of Solutions for a General Linear System . . . . . . . . 84
6.4 Homogeneous Linear Systems. . . . . . . . . . . . . . . . . . . . . . . . . 94
7 Linear Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
7.1 Linear Transformations and Matrices . . . . . . . . . . . . . . . . . . . . 97
7.2 Basic Notions on Maps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
7.3 Kernel and Image of a Linear Map . . . . . . . . . . . . . . . . . . . . . 104
7.4 Isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
7.5 Computing the Kernel of a Linear Map . . . . . . . . . . . . . . . . . . 108
7.6 Computing the Image of a Linear Map . . . . . . . . . . . . . . . . . . 111
7.7 Injectivity and Surjectivity Criteria. . . . . . . . . . . . . . . . . . . . . . 114
7.8 Composition of Linear Maps. . . . . . . . . . . . . . . . . . . . . . . . . . 116
7.9 Change of Basis in a Vector Space . . . . . . . . . . . . . . . . . . . . . 118
8 Dual Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
8.1 The Dual of a Vector Space . . . . . . . . . . . . . . . . . . . . . . . . . . 125
8.2 The Dirac’s Bra-Ket Formalism. . . . . . . . . . . . . . . . . . . . . . . . 128
9 Endomorphisms and Diagonalization . . . . . . . . . . . . . . . . . . . . . . . 131
9.1 Endomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
9.2 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . 133
9.3 The Characteristic Polynomial of an Endomorphism. . . . . . . . . 138
9.4 Diagonalisation of an Endomorphism. . . . . . . . . . . . . . . . . . . . 143
9.5 The Jordan Normal Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
10 Spectral Theorems on Euclidean Spaces . . . . . . . . . . . . . . . . . . . . . 151
10.1 Orthogonal Matrices and Isometries. . . . . . . . . . . . . . . . . . . . . 151
10.2 Self-adjoint Endomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . 156
10.3 Orthogonal Projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
10.4 The Diagonalization of Self-adjoint Endomorphisms. . . . . . . . . 163
10.5 The Diagonalization of Symmetric Matrices . . . . . . . . . . . . . . . 167
11 Rotations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
11.1 Skew-Adjoint Endomorphisms. . . . . . . . . . . . . . . . . . . . . . . . . 173
11.2 The Exponential of a Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 178
11.3 Rotations in Two Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . 180
11.4 Rotations in Three Dimensions . . . . . . . . . . . . . . . . . . . . . . . . 182
11.5 The Lie Algebra soð3Þ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
11.6 The Angular Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
11.7 Rigid Bodies and Inertia Matrix. . . . . . . . . . . . . . . . . . . . . . . . 194
12 Spectral Theorems on Hermitian Spaces. . . . . . . . . . . . . . . . . . . . . 197
12.1 The Adjoint Endomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . 197
12.2 Spectral Theory for Normal Endomorphisms . . . . . . . . . . . . . . 203
12.3 The Unitary Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
13 Quadratic Forms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
13.1 Quadratic Forms on Real Vector Spaces. . . . . . . . . . . . . . . . . . 213
13.2 Quadratic Forms on Complex Vector Spaces . . . . . . . . . . . . . . 222
13.3 The Minkowski Spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
13.4 Electro-Magnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
14 Affine Linear Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
14.1 Affine Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
14.2 Lines and Planes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
14.3 General Linear Affine Varieties and Parallelism . . . . . . . . . . . . 245
14.4 The Cartesian Form of Linear Affine Varieties . . . . . . . . . . . . . 249
14.5 Intersection of Linear Affine Varieties . . . . . . . . . . . . . . . . . . . 258
15 Euclidean Affine Linear Geometry . . . . . . . . . . . . . . . . . . . . . . . . . 269
15.1 Euclidean Affine Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
15.2 Orthogonality Between Linear Affine Varieties. . . . . . . . . . . . . 272
15.3 The Distance Between Linear Affine Varieties . . . . . . . . . . . . . 276
15.4 Bundles of Lines and of Planes . . . . . . . . . . . . . . . . . . . . . . . . 283
15.5 Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
16 Conic Sections. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
16.1 Conic Sections as Geometric Loci . . . . . . . . . . . . . . . . . . . . . . 293
16.2 The Equation of a Conic in Matrix Form . . . . . . . . . . . . . . . . . 298
16.3 Reduction to Canonical Form of a Conic: Translations . . . . . . . 301
16.4 Eccentricity: Part 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307
16.5 Conic Sections and Kepler Motions. . . . . . . . . . . . . . . . . . . . . 309
16.6 Reduction to Canonical Form of a Conic: Rotations . . . . . . . . . 310
16.7 Eccentricity: Part 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318
16.8 Why Conic Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323
Appendix A: Algebraic Structures.. .... .... .... .... .... ..... .... 329
Index .... .... .... .... .... ..... .... .... .... .... .... ..... .... 343
Introduction
Thisbookoriginatesfromacollectionoflecturenotesthatthefirstauthorprepared
at the University of Trieste with Michela Brundu, over a span of fifteen years,
together with the more recent one written by the second author. The notes were
meant for undergraduate classes on linear algebra, geometry and more generally
basic mathematical physics delivered to physics and engineering students, as well
as mathematics students in Italy, Germany and Luxembourg.
Thebookismainlyintendedtobeaself-containedintroductiontothetheoryof
finite-dimensional vector spaces and linear transformations (matrices) with their
spectral analysis both on Euclidean and Hermitian spaces, to affine Euclidean
geometry as well as to quadratic forms and conic sections.
Many topics are introduced and motivated by examples, mostly from physics.
They show how a definition is natural and how the main theorems and results are
first of all plausible before a proof is given. Following this approach, the book
presents a number of examples and exercises, which are meant as a central part in
thedevelopmentofthetheory.Theyareallcompletelysolvedandintendedbothto
guidethestudenttoappreciatetherelevantformalstructuresandtogiveinseveral
cases a proof and a discussion, within a geometric formalism, of results from
physics, notably from mechanics (including celestial) and electromagnetism.
Being the book intended mainly for students in physics and engineering, we
tasked ourselves not to present the mathematical formalism per se. Although we
decided,for clarity'ssakeofourreaders,toorganisethebasicsofthetheoryinthe
classical terms of definitions and the main results as theorems or propositions, we
dooftennotfollowthestandardsequentialformofdefinition—theorem—corollary
—example and provided some two hundred and fifty solved problems given as
exercises.
Chapter 1 of the book presents the Euclidean space used in physics in terms of
applied vectors with respect to orthonormal coordinate system, together with the
operation of scalar, vector and mixed product. They are used both to describe the
motion of a point mass and to introduce the notion of vector field with the most
relevant differential operators acting upon them.
Chapters 2 and 3 are devoted to a general formulation of the theory of
finite-dimensionalvectorspacesequippedwithascalarproduct,whiletheChaps.4
–6 present, via a host of examples and exercises, the theory offinite rank matrices
and their use to solve systems of linear equations.
These are followed by the theory of linear transformations in Chap. 7. Such a
theoryisdescribedinChap.8intermsoftheDirac’sBra-Ketformalism,providing
a link to a geometric–algebraic language used in quantum mechanics.
Thenotionofthediagonalactionofanendomorphismoramatrix(theproblem
ofdiagonalisationandofreductiontotheJordanform)iscentralinthisbook,andit
is introduced in Chap. 9.
Againwithmanysolvedexercisesandexamples,Chap.10describesthespectral
theoryforoperators(matrices)onEuclideanspaces,and(inChap.11)howitallows
onetocharacterisetherotationsinclassicalmechanics.Thisisdonebyintroducing
the Euler angles which parameterise rotations of the physical three-dimensional
space, the notion of angular velocity and by studying the motion of a rigid body
withitsinertiamatrix,andformulatingthedescriptionofthemotionwithrespectto
differentinertialobservers,alsogivingacharacterisationofpolarandaxialvectors.
Chapter 12 is devoted to the spectral theory for matrices acting on Hermitian
spaces in order to present a geometric setting to study a finite level quantum
mechanicalsystem,wherethetimeevolutionisgivenintermsoftheunitarygroup.
All these notions are related with the notion of Lie algebra and to the exponential
map on the space offinite rank matrices.
In Chap. 13, we present the theory of quadratic forms. Our focus is the
description of their transformation properties, so to give the notion of signature,
both in the real and in the complex cases. As the most interesting example of a
non-Euclidean quadratic form, we present the Minkowski spacetime from special
relativity and the Maxwell equations.
In Chaps. 14 and 15, we introduce through many examples the basics of the
Euclideanaffinelineargeometryanddeveloptheminthestudyofconicsections,in
Chap. 16, which are related to the theory of Kepler motions for celestial body in
classicalmechanics.Inparticular,weshowhowtocharacteriseaconicbymeansof
its eccentricity.
A reader of this book is only supposed to know about number sets, more
precisely the natural, integer, rational and real numbers and no additional prior
knowledgeisrequired.Totrytobeasmuchself-containedaspossible,anappendix
collects a few basic algebraic notions, like that of group, ring and field and maps
between them that preserve the structures (homomorphisms), and polynomials in
onevariable.Therearealsoafewbasicpropertiesofthefieldofcomplexnumbers
and of the field of (classes of) integers modulo a prime number.
Trieste, Italy Giovanni Landi
Napoli, Italy Alessandro Zampini
May 2018
Chapter 1
Vectors and Coordinate Systems
Thenotionofavector,ormorepreciselyofavectorappliedatapoint,originatesin
physicswhendealingwithanobservablequantity.Bythisorsimplybyobservable,
onemeansanythingthatcanbemeasuredinthephysicalspace—thespaceofphysical
events— via a suitable measuring process. Examples are the velocity of a point
particle, or its acceleration, or a force acting on it. These are characterised at the
pointofapplicationbyadirection,anorientationandamodulus(ormagnitude).In
the following pages we describe the physical space in terms of points and applied
vectors,andusethesetodescribethephysicalobservablesrelatedtothemotionofa
pointparticlewithrespecttoacoordinatesystem(areferenceframe).Thegeometric
structures introduced in this chapter will be more rigorously analysed in the next
chapters.
1.1 AppliedVectors
We refer to the common intuition of a physical space made of points, where the
notionsofstraightlinebetweentwopointsandofthelengthofasegment(orequiv-
alentlyofdistanceoftwopoints)areassumedtobegiven.Then,avectorv canbe
denotedas
v = B− A or v = AB,
where A,B aretwopointsofthephysicalspace.Then, Aisthepointofapplication
ofv,itsdirectionisthestraightlinejoiningBtoA,itsorientationtheoneofthearrow
pointingfrom Atowards B,anditsmodulustherealnumber(cid:2)B− A(cid:2)=(cid:2)A−B(cid:2),
thatisthelength(withrespecttoafixedunit)ofthesegment AB.
2 1 VectorsandCoordinateSystems
Fig.1.1 Theparallelogramrule
IfS denotestheusualthreedimensionalphysicalspace,wedenoteby
W3 ={B− A| A,B ∈S}
thecollectionofallappliedvectorsatanypointofS andby
V3 ={B− A| B ∈S}
A
thecollectionofallvectorsappliedat AinS.Then
(cid:2)
W3 = V3.
A
A∈S
Remark1.1.1 Oncefixedapoint O inS,oneseesthatthereisabijectionbetween
the set V3 ={B−O | B ∈S} and S itself. Indeed, each point B in S uniquely
O
determines the element B−O in V3, and each element B−O in V3 uniquely
O O
determinesthepoint B inS.
It is well known that the so called parallelogram rule defines in V3 a sum of
O
vectors,where
(A−O)+(B−O)=(C −O),
withC thefourthvertexoftheparallelogramwhoseotherthreeverticesare A, O,
B,asshowninFig.1.1.
The vector 0= O−O is called the zero vector (or null vector); notice that its
modulusiszero,whileitsdirectionandorientationareundefined.
It is evident that V3 is closed with respect to the notion of sum defined above.
O
Thatsuchasumisassociativeandabelianispartofthecontentoftheproposition
thatfollows.
Proposition1.1.2 Thedatum(V3,+,0)isanabeliangroup.
O
Proof Clearlythezerovector0istheneutral(identity)elementforthesuminV3,
O
thataddedtoanyvectorleavethelatterunchanged.AnyvectorA−Ohasaninverse
1.1 AppliedVectors 3
Fig.1.2 Theoppositeofavector:A(cid:4)−O =−(A−O)
Fig.1.3 Theassociativityofthevectorsum
withrespecttothesum(thatis,anyvectorhasanoppositevector)givenby A(cid:4)−O,
where A(cid:4) isthesymmetricpointto Awithrespectto O onthestraightlinejoining
Ato O (seeFig.1.2).
From its definition the sum of two vectors is a commutative operation. For the
associativitywegiveapictorialargumentinFig.1.3. (cid:2)
There is indeed more structure. The physical intuition allows one to consider
multiplesofanappliedvector.ConcerningthecollectionV3,thisamountstodefine
O
anoperationinvolvingvectorsappliedinOandrealnumbers,which,inordernotto
createconfusionwithvectors,arecalled(real)scalars.
Definition1.1.3 Giventhescalarλ∈ Randthevector A−O ∈ V3,theproduct
O
byascalar
B − O = λ(A − O)
isthevectorsuchthat:
(i) A,B,O areonthesame(straight)line,
(ii) B − O and A − O have the same orientation if λ>0, while A − O and
B − O haveoppositeorientationsifλ<0,
(iii) (cid:2)B − O(cid:2) = |λ|(cid:2)A − O(cid:2).
The main properties of the operation of product by a scalar are given in the
followingproposition.
Proposition1.1.4 For any pair of scalars λ,μ ∈ R and any pair of vectors
A − O, B − O ∈ V3,itholdsthat:
O