Linear Algebra and Analytic Geometry for Physical Sciences

Undergraduate Lecture Notes in Physics

Giovanni Landi · Alessandro Zampini

Linear Algebra

and Analytic

Geometry

for Physical

Sciences

Undergraduate Lecture Notes in Physics

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Giovanni Landi • Alessandro Zampini

Linear Algebra and Analytic

Geometry for Physical

Sciences

123

Giovanni Landi

University of Trieste

Trieste

Italy

Alessandro Zampini

INFN Sezione di Napoli

Napoli

Italy

ISSN 2192-4791 ISSN 2192-4805 (electronic)

Undergraduate Lecture Notes in Physics

ISBN 978-3-319-78360-4 ISBN 978-3-319-78361-1 (eBook)

https://doi.org/10.1007/978-3-319-78361-1

Library of Congress Control Number: 2018935878

© Springer International Publishing AG, part of Springer Nature 2018

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To our families

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

vii

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

viii Contents

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

Contents ix

Introduction

This book originates from a collection of lecture notes that the first author prepared

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.

The book is mainly intended to be a self-contained introduction to the theory of

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

the development of the theory. They are all completely solved and intended both to

guide the student to appreciate the relevant formal structures and to give in several

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's sake of our readers, to organise the basics of the theory in the

classical terms of definitions and the main results as theorems or propositions, we

do often not follow the standard sequential form of definition—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.

xi

Chapters 2 and 3 are devoted to a general formulation of the theory of

finite-dimensional vector spaces equipped with a scalar product, while the Chaps. 4

–6 present, via a host of examples and exercises, the theory of finite 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

theory is described in Chap. 8 in terms of the Dirac’s Bra-Ket formalism, providing

a link to a geometric–algebraic language used in quantum mechanics.

The notion of the diagonal action of an endomorphism or a matrix (the problem

of diagonalisation and of reduction to the Jordan form) is central in this book, and it

is introduced in Chap. 9.

Again with many solved exercises and examples, Chap. 10 describes the spectral

theory for operators (matrices) on Euclidean spaces, and (in Chap. 11) how it allows

one to characterise the rotations in classical mechanics. This is done by introducing

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

with its inertia matrix, and formulating the description of the motion with respect to

different inertial observers, also giving a characterisation of polar and axial vectors.

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

mechanical system, where the time evolution is given in terms of the unitary group.

All these notions are related with the notion of Lie algebra and to the exponential

map on the space of finite 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

Euclidean affine linear geometry and develop them in the study of conic sections, in

Chap. 16, which are related to the theory of Kepler motions for celestial body in

classical mechanics. In particular, we show how to characterise a conic by means of

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

knowledge is required. To try to be as much self-contained as possible, an appendix

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

one variable. There are also a few basic properties of the field of complex numbers

and of the field of (classes of) integers modulo a prime number.

Giovanni Landi

Alessandro Zampini

Trieste, Italy

Napoli, Italy

May 2018

xii Introduction

Chapter 1

Vectors and Coordinate Systems

The notion of a vector, or more precisely of a vector applied at a point, originates in

physics when dealing with an observable quantity. By this or simply by observable,

one means anything that can be measured in the physical space—the space of physical

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

point of application by a direction, an orientation and a modulus (or magnitude). In

the following pages we describe the physical space in terms of points and applied

vectors, and use these to describe the physical observables related to the motion of a

point particle with respect to a coordinate system (a reference frame). The geometric

structures introduced in this chapter will be more rigorously analysed in the next

chapters.

1.1 Applied Vectors

We refer to the common intuition of a physical space made of points, where the

notions of straight line between two points and of the length of a segment (or equiv￾alently of distance of two points) are assumed to be given. Then, a vector v can be

denoted as

v = B − A or v = AB,

where A, B are two points of the physical space. Then, A is the point of application

of v, its direction is the straight line joining B to A, its orientation the one of the arrow

pointing from A towards B, and its modulus the real number B − A=A − B,

that is the length (with respect to a fixed unit) of the segment AB.

© Springer International Publishing AG, part of Springer Nature 2018

G. Landi and A. Zampini, Linear Algebra and Analytic Geometry

for Physical Sciences, Undergraduate Lecture Notes in Physics,

https://doi.org/10.1007/978-3-319-78361-1_1

1

2 1 Vectors and Coordinate Systems

Fig. 1.1 The parallelogram rule

If S denotes the usual three dimensional physical space, we denote by

W3 = {B − A | A, B ∈ S}

the collection of all applied vectors at any point of S and by

V3

A = {B − A | B ∈ S}

the collection of all vectors applied at A in S. Then

W3 =

A∈S

V3

A.

Remark 1.1.1 Once fixed a point O in S, one sees that there is a bijection between

the set V3

O = {B − O | B ∈ S} and S itself. Indeed, each point B in S uniquely

determines the element B − O in V3

O , and each element B − O in V3

O uniquely

determines the point B in S.

It is well known that the so called parallelogram rule defines in V3

O a sum of

vectors, where

(A − O) + (B − O) = (C − O),

with C the fourth vertex of the parallelogram whose other three vertices are A, O,

B, as shown in Fig. 1.1.

The vector 0 = O − O is called the zero vector (or null vector); notice that its

modulus is zero, while its direction and orientation are undefined.

It is evident that V3

O is closed with respect to the notion of sum defined above.

That such a sum is associative and abelian is part of the content of the proposition

that follows.

Proposition 1.1.2 The datum (V3

O , +, 0) is an abelian group.

Proof Clearly the zero vector 0 is the neutral (identity) element for the sum in V3

O ,

that added to any vector leave the latter unchanged. Any vector A − O has an inverse

1.1 Applied Vectors 3

Fig. 1.2 The opposite of a vector: A



− O = −(A − O)

Fig. 1.3 The associativity of the vector sum

with respect to the sum (that is, any vector has an opposite vector) given by A − O,

where A is the symmetric point to A with respect to O on the straight line joining

A to O (see Fig. 1.2).

From its definition the sum of two vectors is a commutative operation. For the

associativity we give a pictorial argument in Fig. 1.3.

There is indeed more structure. The physical intuition allows one to consider

multiples of an applied vector. Concerning the collection V3

O , this amounts to define

an operation involving vectors applied in O and real numbers, which, in order not to

create confusion with vectors, are called (real) scalars.

Definition 1.1.3 Given the scalar λ ∈ R and the vector A − O ∈ V3

O , the product

by a scalar

B − O = λ(A − O)

is the vector such that:

(i) A, B, O are on the same (straight) line,

(ii) B − O and A − O have the same orientation if λ > 0, while A − O and

B − O have opposite orientations if λ < 0,

(iii) B − O=|λ| A − O.

The main properties of the operation of product by a scalar are given in the

following proposition.

Proposition 1.1.4 For any pair of scalars λ, μ ∈ R and any pair of vectors

A − O, B − O ∈ V3

O , it holds that:

4 1 Vectors and Coordinate Systems

Fig. 1.4 The scaling λ(C − O) = (C − O) with λ > 1

1. λ(μ(A − O)) = (λμ)(A − O),

2. 1(A − O) = A − O,

3. λ ((A − O) + (B − O)) = λ(A − O) + λ(B − O),

4. (λ + μ)(A − O) = λ(A − O) + μ(A − O).

Proof 1. Set C − O = λ (μ(A − O)) and D − O = (λμ)(A − O). If one of

the scalars λ, μ is zero, one trivially has C − O = 0 and D − O = 0, so

Point 1. is satisfied. Assume now that λ = 0 and μ = 0. Since, by definition,

both C and D are points on the line determined by O and A, the vectors C − O

and D − O have the same direction. It is easy to see that C − O and D − O

have the same orientation: it will coincide with the orientation of A − O or not,

depending on the sign of the product λμ = 0. Since |λμ|=|λ||μ| ∈ R, one has

C − O=D − O.

2. It follows directly from the definition.

3. SetC − O = (A − O) + (B − O) andC − O = (A − O) + (B − O),

with A − O = λ(A − O) and B − O = λ(B − O).

We verify that λ(C − O) = C − O (see Fig. 1.4).

Since O A is parallel to O A by definition, then BC is parallel to B

C

; O B is

indeed parallel to O B

, so that the planar anglesOBC and O B

C are equal.

Also λ(O B) = O B

, λ(O A) = O A

, and λ(BC) = B

C

. It follows that the

triangles OBC and O B

C are similar: the vector OC is then parallel OC and

they have the same orientation, with OC

= λ OC. From this we obtain

OC = λ(OC).

4. The proof is analogue to the one in point 3.

What we have described above shows that the operations of sum and product by a

scalar give V3

O an algebraic structure which is richer than that of abelian group. Such

a structure, that we shall study in detail in Chap. 2, is called in a natural way vector

space.