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Linear Algebra I
Linear Algebra I Martin Otto Winter Term 2013/14 Contents 1 Introduction7 1.1 Motivating Examples.......................7 1.1.1 The two-dimensional real plane.............7 1.1.2 Three-dimensional real space............... 14 1.1.3 Systems of linear equations over Rn ........... 15 1.1.4 Linear spaces over Z2 ................... 21 1.2 Basics, Notation and Conventions................ 27 1.2.1 Sets............................ 27 1.2.2 Functions......................... 29 1.2.3 Relations......................... 34 1.2.4 Summations........................ 36 1.2.5 Propositional logic.................... 36 1.2.6 Some common proof patterns.............. 37 1.3 Algebraic Structures....................... 39 1.3.1 Binary operations on a set................ 39 1.3.2 Groups........................... 40 1.3.3 Rings and fields...................... 42 1.3.4 Aside: isomorphisms of algebraic structures...... 44 2 Vector Spaces 47 2.1 Vector spaces over arbitrary fields................ 47 2.1.1 The axioms........................ 48 2.1.2 Examples old and new.................. 50 2.2 Subspaces............................. 53 2.2.1 Linear subspaces..................... 53 2.2.2 Affine subspaces...................... 56 2.3 Aside: affine and linear spaces.................. 58 2.4 Linear dependence and independence.............. 60 3 4 Linear Algebra I | Martin Otto 2013 2.4.1 Linear combinations and spans............. 60 2.4.2 Linear (in)dependence.................. 62 2.5 Bases and dimension....................... 65 2.5.1 Bases............................ 65 2.5.2 Finite-dimensional vector spaces............. 66 2.5.3 Dimensions of linear and affine subspaces........ 71 2.5.4 Existence of bases..................... 72 2.6 Products, sums and quotients of spaces............. 73 2.6.1 Direct products...................... 73 2.6.2 Direct sums of subspaces................ -
Vector Spaces in Physics
San Francisco State University Department of Physics and Astronomy August 6, 2015 Vector Spaces in Physics Notes for Ph 385: Introduction to Theoretical Physics I R. Bland TABLE OF CONTENTS Chapter I. Vectors A. The displacement vector. B. Vector addition. C. Vector products. 1. The scalar product. 2. The vector product. D. Vectors in terms of components. E. Algebraic properties of vectors. 1. Equality. 2. Vector Addition. 3. Multiplication of a vector by a scalar. 4. The zero vector. 5. The negative of a vector. 6. Subtraction of vectors. 7. Algebraic properties of vector addition. F. Properties of a vector space. G. Metric spaces and the scalar product. 1. The scalar product. 2. Definition of a metric space. H. The vector product. I. Dimensionality of a vector space and linear independence. J. Components in a rotated coordinate system. K. Other vector quantities. Chapter 2. The special symbols ij and ijk, the Einstein summation convention, and some group theory. A. The Kronecker delta symbol, ij B. The Einstein summation convention. C. The Levi-Civita totally antisymmetric tensor. Groups. The permutation group. The Levi-Civita symbol. D. The cross Product. E. The triple scalar product. F. The triple vector product. The epsilon killer. Chapter 3. Linear equations and matrices. A. Linear independence of vectors. B. Definition of a matrix. C. The transpose of a matrix. D. The trace of a matrix. E. Addition of matrices and multiplication of a matrix by a scalar. F. Matrix multiplication. G. Properties of matrix multiplication. H. The unit matrix I. Square matrices as members of a group. -
Multivector Differentiation and Linear Algebra 0.5Cm 17Th Santaló
Multivector differentiation and Linear Algebra 17th Santalo´ Summer School 2016, Santander Joan Lasenby Signal Processing Group, Engineering Department, Cambridge, UK and Trinity College Cambridge [email protected], www-sigproc.eng.cam.ac.uk/ s jl 23 August 2016 1 / 78 Examples of differentiation wrt multivectors. Linear Algebra: matrices and tensors as linear functions mapping between elements of the algebra. Functional Differentiation: very briefly... Summary Overview The Multivector Derivative. 2 / 78 Linear Algebra: matrices and tensors as linear functions mapping between elements of the algebra. Functional Differentiation: very briefly... Summary Overview The Multivector Derivative. Examples of differentiation wrt multivectors. 3 / 78 Functional Differentiation: very briefly... Summary Overview The Multivector Derivative. Examples of differentiation wrt multivectors. Linear Algebra: matrices and tensors as linear functions mapping between elements of the algebra. 4 / 78 Summary Overview The Multivector Derivative. Examples of differentiation wrt multivectors. Linear Algebra: matrices and tensors as linear functions mapping between elements of the algebra. Functional Differentiation: very briefly... 5 / 78 Overview The Multivector Derivative. Examples of differentiation wrt multivectors. Linear Algebra: matrices and tensors as linear functions mapping between elements of the algebra. Functional Differentiation: very briefly... Summary 6 / 78 We now want to generalise this idea to enable us to find the derivative of F(X), in the A ‘direction’ – where X is a general mixed grade multivector (so F(X) is a general multivector valued function of X). Let us use ∗ to denote taking the scalar part, ie P ∗ Q ≡ hPQi. Then, provided A has same grades as X, it makes sense to define: F(X + tA) − F(X) A ∗ ¶XF(X) = lim t!0 t The Multivector Derivative Recall our definition of the directional derivative in the a direction F(x + ea) − F(x) a·r F(x) = lim e!0 e 7 / 78 Let us use ∗ to denote taking the scalar part, ie P ∗ Q ≡ hPQi. -
Appendix A: Matrices and Tensors
Appendix A: Matrices and Tensors A.1 Introduction and Rationale The purpose of this appendix is to present the notation and most of the mathematical techniques that will be used in the body of the text. The audience is assumed to have been through several years of college level mathematics that included the differential and integral calculus, differential equations, functions of several variables, partial derivatives, and an introduction to linear algebra. Matrices are reviewed briefly and determinants, vectors, and tensors of order two are described. The application of this linear algebra to material that appears in undergraduate engineering courses on mechanics is illustrated by discussions of concepts like the area and mass moments of inertia, Mohr’s circles and the vector cross and triple scalar products. The solutions to ordinary differential equations are reviewed in the last two sections. The notation, as far as possible, will be a matrix notation that is easily entered into existing symbolic computational programs like Maple, Mathematica, Matlab, and Mathcad etc. The desire to represent the components of three-dimensional fourth order tensors that appear in anisotropic elasticity as the components of six-dimensional second order tensors and thus represent these components in matrices of tensor components in six dimensions leads to the nontraditional part of this appendix. This is also one of the nontraditional aspects in the text of the book, but a minor one. This is described in Sect. A.11, along with the rationale for this approach. A.2 Definition of Square, Column, and Row Matrices An r by c matrix M is a rectangular array of numbers consisting of r rows and c columns, S.C. -
An Attempt to Intuitively Introduce the Dot, Wedge, Cross, and Geometric Products
An attempt to intuitively introduce the dot, wedge, cross, and geometric products Peeter Joot March 21, 2008 1 Motivation. Both the NFCM and GAFP books have axiomatic introductions of the gener- alized (vector, blade) dot and wedge products, but there are elements of both that I was unsatisfied with. Perhaps the biggest issue with both is that they aren’t presented in a dumb enough fashion. NFCM presents but does not prove the generalized dot and wedge product operations in terms of symmetric and antisymmetric sums, but it is really the grade operation that is fundamental. You need that to define the dot product of two bivectors for example. GAFP axiomatic presentation is much clearer, but the definition of general- ized wedge product as the totally antisymmetric sum is a bit strange when all the differential forms book give such a different definition. Here I collect some of my notes on how one starts with the geometric prod- uct action on colinear and perpendicular vectors and gets the familiar results for two and three vector products. I may not try to generalize this, but just want to see things presented in a fashion that makes sense to me. 2 Introduction. The aim of this document is to introduce a “new” powerful vector multiplica- tion operation, the geometric product, to a student with some traditional vector algebra background. The geometric product, also called the Clifford product 1, has remained a relatively obscure mathematical subject. This operation actually makes a great deal of vector manipulation simpler than possible with the traditional methods, and provides a way to naturally expresses many geometric concepts. -
Dirac Notation 1 Vectors
Physics 324, Fall 2001 Dirac Notation 1 Vectors 1.1 Inner product T Recall from linear algebra: we can represent a vector V as a column vector; then V y = (V )∗ is a row vector, and the inner product (another name for dot product) between two vectors is written as AyB = A∗B1 + A∗B2 + : (1) 1 2 ··· In conventional vector notation, the above is just A~∗ B~ . Note that the inner product of a vector with itself is positive definite; we can define the· norm of a vector to be V = pV V; (2) j j y which is a non-negative real number. (In conventional vector notation, this is V~ , which j j is the length of V~ ). 1.2 Basis vectors We can expand a vector in a set of basis vectors e^i , provided the set is complete, which means that the basis vectors span the whole vectorf space.g The basis is called orthonormal if they satisfy e^iye^j = δij (orthonormality); (3) and an orthonormal basis is complete if they satisfy e^ e^y = I (completeness); (4) X i i i where I is the unit matrix. (note that a column vectore ^i times a row vectore ^iy is a square matrix, following the usual definition of matrix multiplication). Assuming we have a complete orthonormal basis, we can write V = IV = e^ e^yV V e^ ;V (^eyV ) : (5) X i i X i i i i i ≡ i ≡ The Vi are complex numbers; we say that Vi are the components of V in the e^i basis. -
Appendix a Multilinear Algebra and Index Notation
Appendix A Multilinear algebra and index notation Contents A.1 Vector spaces . 164 A.2 Bases, indices and the summation convention 166 A.3 Dual spaces . 169 A.4 Inner products . 170 A.5 Direct sums . 174 A.6 Tensors and multilinear maps . 175 A.7 The tensor product . 179 A.8 Symmetric and exterior algebras . 182 A.9 Duality and the Hodge star . 188 A.10 Tensors on manifolds . 190 If linear algebra is the study of vector spaces and linear maps, then multilinear algebra is the study of tensor products and the natural gener- alizations of linear maps that arise from this construction. Such concepts are extremely useful in differential geometry but are essentially algebraic rather than geometric; we shall thus introduce them in this appendix us- ing only algebraic notions. We'll see finally in A.10 how to apply them to tangent spaces on manifolds and thus recoverx the usual formalism of tensor fields and differential forms. Along the way, we will explain the conventions of \upper" and \lower" index notation and the Einstein sum- mation convention, which are standard among physicists but less familiar in general to mathematicians. 163 164 APPENDIX A. MULTILINEAR ALGEBRA A.1 Vector spaces and linear maps We assume the reader is somewhat familiar with linear algebra, so at least most of this section should be review|its main purpose is to establish notation that is used in the rest of the notes, as well as to clarify the relationship between real and complex vector spaces. Throughout this appendix, let F denote either of the fields R or C; we will refer to elements of this field as scalars. -
1 Sets and Set Notation. Definition 1 (Naive Definition of a Set)
LINEAR ALGEBRA MATH 2700.006 SPRING 2013 (COHEN) LECTURE NOTES 1 Sets and Set Notation. Definition 1 (Naive Definition of a Set). A set is any collection of objects, called the elements of that set. We will most often name sets using capital letters, like A, B, X, Y , etc., while the elements of a set will usually be given lower-case letters, like x, y, z, v, etc. Two sets X and Y are called equal if X and Y consist of exactly the same elements. In this case we write X = Y . Example 1 (Examples of Sets). (1) Let X be the collection of all integers greater than or equal to 5 and strictly less than 10. Then X is a set, and we may write: X = f5; 6; 7; 8; 9g The above notation is an example of a set being described explicitly, i.e. just by listing out all of its elements. The set brackets {· · ·} indicate that we are talking about a set and not a number, sequence, or other mathematical object. (2) Let E be the set of all even natural numbers. We may write: E = f0; 2; 4; 6; 8; :::g This is an example of an explicity described set with infinitely many elements. The ellipsis (:::) in the above notation is used somewhat informally, but in this case its meaning, that we should \continue counting forever," is clear from the context. (3) Let Y be the collection of all real numbers greater than or equal to 5 and strictly less than 10. Recalling notation from previous math courses, we may write: Y = [5; 10) This is an example of using interval notation to describe a set. -
Vectors and Matrices Notes
Vectors and Matrices Notes. Jonathan Coulthard [email protected] 1 Index Notation Index notation may seem quite intimidating at first, but once you get used to it, it will allow us to prove some very tricky vector and matrix identities with very little effort. As with most things, it will only become clearer with practice, and so it is a good idea to work through the examples for yourself, and try out some of the exercises. Example: Scalar Product Let's start off with the simplest possible example: the dot product. For real column vectors a and b, 0 1 b1 Bb2C a · b = aT b = a a a ··· B C = a b + a b + a b + ··· (1) 1 2 3 Bb3C 1 1 2 2 3 3 @ . A . or, written in a more compact notation X a · b = aibi; (2) i where the σ means that we sum over all values of i. Example: Matrix-Column Vector Product Now let's take matrix-column vector multiplication, Ax = b. 2 3 2 3 2 3 A11 A12 A23 ··· x1 b1 6A21 A22 A23 ··· 7 6x27 6b27 6 7 6 7 = 6 7 (3) 6A31 A32 A33 ···7 6x37 6b37 4 . 5 4 . 5 4 . 5 . .. You are probably used to multiplying matrices by visualising multiplying the elements high- lighted in the red boxes. Written out explicitly, this is b2 = A21x1 + A22x2 + A23x3 + ··· (4) If we were to shift the A box and the b box down one place, we would instead get b3 = A31x1 + A32x2 + A33x3 + ··· (5) 1 It should be clear then, that in general, for the ith element of b, we can write bi = Ai1x1 + Ai2x2 + Ai3x3 + ··· (6) Or, in our more compact notation, X bi = Aijxj: (7) j Note that if the matrix A had only one column, then i would take only one value (i = 1). -
Low-Level Image Processing with the Structure Multivector
Low-Level Image Processing with the Structure Multivector Michael Felsberg Bericht Nr. 0202 Institut f¨ur Informatik und Praktische Mathematik der Christian-Albrechts-Universitat¨ zu Kiel Olshausenstr. 40 D – 24098 Kiel e-mail: [email protected] 12. Marz¨ 2002 Dieser Bericht enthalt¨ die Dissertation des Verfassers 1. Gutachter Prof. G. Sommer (Kiel) 2. Gutachter Prof. U. Heute (Kiel) 3. Gutachter Prof. J. J. Koenderink (Utrecht) Datum der mundlichen¨ Prufung:¨ 12.2.2002 To Regina ABSTRACT The present thesis deals with two-dimensional signal processing for computer vi- sion. The main topic is the development of a sophisticated generalization of the one-dimensional analytic signal to two dimensions. Motivated by the fundamental property of the latter, the invariance – equivariance constraint, and by its relation to complex analysis and potential theory, a two-dimensional approach is derived. This method is called the monogenic signal and it is based on the Riesz transform instead of the Hilbert transform. By means of this linear approach it is possible to estimate the local orientation and the local phase of signals which are projections of one-dimensional functions to two dimensions. For general two-dimensional signals, however, the monogenic signal has to be further extended, yielding the structure multivector. The latter approach combines the ideas of the structure tensor and the quaternionic analytic signal. A rich feature set can be extracted from the structure multivector, which contains measures for local amplitudes, the local anisotropy, the local orientation, and two local phases. Both, the monogenic signal and the struc- ture multivector are combined with an appropriate scale-space approach, resulting in generalized quadrature filters. -
Appendix: an Index Notation for Multivariate Statistical Analysis
19 : APPENDIX An Index Notation for Multivariate Analysis 1. Tensor Products and Formal Orderings The algebra of multivariate statistical analysis is, predominantly, the al- gebra of tensor products, of which the Kronecker product of matrices is a particular instance. The Kronecker product of the t × n matrix B =[blj] and the s × m matrix A =[aki] is defined as an ts×nm matrix B ⊗A =[bljA] whose ljth partition is bljA. In many statistical texts, this definition provides the basis for subsequent alge- braic developments. The disadvantage of using the definition without support from the theory of multilinear algebra is that the results which are generated often seem purely technical and lacking in intuitive appeal. Another approach to the algebra of tensor products relies upon the algebra of abstract vector spaces. Thus The tensor product U⊗Vof two finite-dimensional vector spaces U and V may be defined as the dual of the vector space of all bilinear functionals on U and V. This definition facilitates the development of a rigorous abstract theory. In particular, U⊗V, defined in this way, already has all the features of a vector space. However, the definition also leads to acute technical difficulties when we seek to represent the resulting algebra in terms of coordinate vectors and matrices. The approach which we shall adopt here is to define tensor products in terms of formal products. According to this approach, a definition of U⊗V may ⊗ be obtained by considering the set of all objects of the form i j xij(ui vj), where ui ⊗vj, which is described as an elementary or decomposable tensor prod- uct, comprises an ordered pair of elements taken from the two vector spaces. -
Relativistic Inversion, Invariance and Inter-Action
S S symmetry Article Relativistic Inversion, Invariance and Inter-Action Martin B. van der Mark †,‡ and John G. Williamson *,‡ The Quantum Bicycle Society, 12 Crossburn Terrace, Troon KA1 07HB, Scotland, UK; [email protected] * Correspondence: [email protected] † Formerly of Philips Research, 5656 AE Eindhoven, The Netherlands. ‡ These authors contributed equally to this work. Abstract: A general formula for inversion in a relativistic Clifford–Dirac algebra has been derived. Identifying the base elements of the algebra as those of space and time, the first order differential equations over all quantities proves to encompass the Maxwell equations, leads to a natural extension incorporating rest mass and spin, and allows an integration with relativistic quantum mechanics. Although the algebra is not a division algebra, it parallels reality well: where division is undefined turns out to correspond to physical limits, such as that of the light cone. The divisor corresponds to invariants of dynamical significance, such as the invariant interval, the general invariant quantities in electromagnetism, and the basis set of quantities in the Dirac equation. It is speculated that the apparent 3-dimensionality of nature arises from a beautiful symmetry between the three-vector algebra and each of four sets of three derived spaces in the full 4-dimensional algebra. It is conjectured that elements of inversion may play a role in the interaction of fields and matter. Keywords: invariants; inversion; division; non-division algebra; Dirac algebra; Clifford algebra; geometric algebra; special relativity; photon interaction Citation: van der Mark, M.B.; 1. Introduction Williamson, J.G. Relativistic Inversion, Invariance and Inter-Action.