DOCSLIB.ORG

Linear Algebra II

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Linear Algebra II Linear Algebra II Martin Otto Summer Term 2018 Contents 1 Eigenvalues and Diagonalisation7 1.1 Eigenvectors and eigenvalues................... 10 1.2 Polynomials............................ 16 1.2.1 Algebra of polynomials.................. 17 1.2.2 Division of polynomials.................. 20 1.2.3 Polynomials over the real and complex numbers.... 24 1.3 Upper triangle form........................ 25 1.4 The Cayley{Hamilton Theorem................. 27 1.5 Minimal polynomial and diagonalisation............ 32 1.6 Jordan Normal Form....................... 36 1.6.1 Block decomposition, part 1............... 37 1.6.2 Block decomposition, part 2............... 39 1.6.3 Jordan normal form.................... 45 2 Euclidean and Unitary Spaces 49 2.1 Euclidean and unitary vector spaces............... 51 2.1.1 The standard scalar products in Rn and Cn ...... 51 2.1.2 Cn as a unitary space................... 53 2.1.3 Bilinear and semi-bilinear forms............. 55 2.1.4 Scalar products in euclidean and unitary spaces.... 58 2.2 Further examples......................... 62 2.3 Orthogonality and orthonormal bases.............. 64 2.3.1 Orthonormal bases.................... 64 2.3.2 Orthogonality and orthogonal complements...... 66 2.3.3 Orthogonal and unitary maps.............. 69 2.4 Endomorphisms in euclidean or unitary spaces......... 74 2.4.1 The adjoint map..................... 75 2.4.2 Diagonalisation of self-adjoint maps and matrices... 76 3 4 Linear Algebra II | Martin Otto 2018 2.4.3 Normal maps and matrices................ 78 3 Bilinear and Quadratic Forms 81 3.1 Matrix representations of bilinear forms............. 81 3.2 Simultaneous diagonalisation................... 82 3.2.1 Symmetric bilinear forms vs. self-adjoint maps..... 83 3.2.2 Principal axes....................... 84 3.2.3 Positive definiteness................... 88 3.3 Quadratic forms and quadrics.................. 89 3.3.1 Quadrics in Rn ...................... 92 3.3.2 Projective space Pn .................... 96 3.3.3 Appendix: an example from classical mechanics.... 100 LA II | Martin Otto 2018 5 Notational conventions In this second part, we adopt the following conventions and notational sim- plifications over and above conventions established in part I. • all bases are understood to be labelled bases, with individual basis vectors listed as for instance in B = (b1;:::; bn). • we extend the idea of summation notation to products, writing for instance n Y λi for λ1 · ::: · λn: i=1 and also to sums and direct sums of subspaces, as in k X Ui for U1 + ··· + Uk i=1 and k M Ui for U1 ⊕ · · · ⊕ Uk: i=1 • in rings (like Hom(V; V )) we use exponentiation notation for iterated products, always identifying the power to exponent zero with the unit element (neutral element w.r.t. multiplication). For instance, if ' is an endomorphism of V , we write k 0 ' for ' ◦ · · · ◦ ' where ' = idV : | {z } k times 6 Linear Algebra II | Martin Otto 2018 Chapter 1 Eigenvalues and Diagonalisation One of the core topics of linear algebra concerns the choice of suitable bases for the representation of linear maps. For an endomorphism ': V ! V of the n-dimensional F-vector space V , we want to find a basis B = (b1;:::; bn) of B V that is specially adapted to make the matrix representation A' = [[']]B 2 F(n;n) as simple as possible for the analysis of the map ' itself. This chapter starts from the study of eigenvectors of '. Such an eigenvector is a vector v 6= 0 that is mapped to a scalar multiple of itself under '. So '(v) = λv for some λ 2 F, the corresponding eigenvalue. If v is an eigenvector (with eigenvalue λ), it spans a one-dimensional subspace U = fµv: µ 2 Fg ⊆ V , which is invariant under ' (or preserved by ') in the sense that '(u) = λu 2 U for all u 2 U. In other words, in the direction of v, ' acts as a rescaling with factor λ. But for instance over the standard two-dimensional R-vector space R2, an endomorphism need not have any eigenvectors. Consider the example of a rotation through an angle 0 < α < π; this linear map preserves no 1-dimensional subspaces at all. In contrast, we shall see later that any en- domorphism of R3 must have at least one eigenvector. In the case of a non- trivial rotation, for instance, the axis of rotation gives rise to an eigenvector with eigenvalue 1. In fact eigenvalues and eigenvectors often have further significance, either geometrically or in terms of the phenomena modelled by a linear map. To give an example, consider the homogeneous linear differential equation of the 7 8 Linear Algebra II | Martin Otto 2018 harmonic oscillator d2 f(t) + cf(t) = 0 dt2 with a positive constant c 2 R and for C1 functions f : R ! R (modelling, for instance, the position of a mass attached to a spring as a function of time t). One may regard this as the problem of finding the eigenvectors d2 1 for eigenvalue −c of the linear operator dt2 that maps a C function to its second derivative.p In this case, therep are two linearly independent solutions f(t) = sin( c t) and f(t) = cos( c t) which span the solution space of this differential equation. The eigenvalue −c of d2 that we look at here is related dtp2 to the frequency of the oscillation (which is c=2π). In quantum mechanics, states of a physical system are modelled as vectors of a C-vector space (e.g., of wave functions). Associated physical quantities (observables) are described by linear (differential) operators on such states, which are endomorphisms of the state space. The eigenvectors of these op- erators are the possible values for measurements of those observables. Here one seeks bases of the state space made up from eigenvectors (eigenstates) associated with particular values for the quantity under consideration via their eigenvalues. W.r.t. such a basis an arbitrary state can be represented as a linear combination (superposition) of eigenstates, accounting for com- ponents which each have their definite value for that observable, but mixed in a composite state with different possible outcomes for its measurement. If V has a basis consisting of eigenvectors of an endomorphism ': V ! V , then w.r.t. that basis, ' is represented by a diagonal matrix, with the eigenvalues as entries on the diagonal, and all other entries equal to 0. Matrix arithmetic is often much simpler for diagonal matrices, and it therefore makes sense to apply a corresponding basis transformation to achieve diagonal form where this is possible. Example 1.0.1 Consider for instance a square matrix A 2 R(n;n) (or C(n;n)). Suppose that there is a regular matrix C (for a basis transformation) such that 0 1 λ1 0 ::: 0 B 0 λ 0 C ^ −1 B 2 C A = CAC = B . .. C @ . 0 A 0 ··· 0 λn LA II | Martin Otto 2018 9 is a diagonal matrix. Suppose now that we want to evaluate powers of A: 0 1 2 A = En, A = A, A = AA, . We note that the corresponding powers of A^ are very easily computed. One checks that 0 ` 1 λ1 0 ::: 0 B 0 λ` 0 C ^` B 2 C A = B . .. C : @ . 0 A ` 0 ··· 0 λn Then A` = (C−1AC^ )` = (C−1AC^ ) · ::: · (C−1AC^ ) = C−1A^`C | {z } ` times is best computed via the detour through A^. Example 1.0.2 Consider the linear differential equation d v(t) = Av(t) dt for a vector-valued C1 function v: t 7! v(t) 2 Rn, where the matrix A 2 R(n;n) is fixed. In close analogy with the one-dimensional case, one can find a solution using the exponential function. Here the exponential function has to be considered as a matrix-valued function on matrices, B 7! eB, defined by tA its series expansion. The function t 7! v(t) := e v0 solves the differential tA equation for initial value v(0) = v0. Here e stands for the series 1 X tkAk 2 (n;n); k! R k=0 tA which can be shown to converge for all t. The value v(t) = e v0 is the result tA of applying the matrix e to the vector v0. If it so happens that there is a basis relative to which A is similar to a diagonal matrix A^ = CAC−1, as in the previous example, then we may solve d ^ the differential equation dt v^(t) = Av^(t) instead, and merely have to express the initial value v0 in the new basis as v^0 = Cv0. The evaluation of the exponential function on the diagonal matrix A is easy (as in the previous example): 0 1 0 tλ1 1 λ1 0 ::: 0 e 0 ::: 0 B 0 λ 0 C B 0 etλ2 0 C ^ B 2 C tA^ B C A = B . .. C ) e = B . .. C : @ . 0 A @ . 0 A tλn 0 ··· 0 λn 0 ··· 0 e 10 Linear Algebra II | Martin Otto 2018 We correspondingly find that the solution (in the adapted basis) is tA^ v^(t) = e v^0; which then transforms back into the original basis according to −1 tA^ v(t) = C e C]v0: Convention: in this chapter, unless otherwise noted, we fix a finite-dimen- sional F-vector space V of dimension greater than 0 throughout. We shall occasionally look at specific examples and specify concrete fields F. 1.1 Eigenvectors and eigenvalues Recall from chapter 3 in part I that Hom(V; V ) stands for the space of all linear maps ': V ! V (vector space homomorphisms from V to V , i.e., endomorphisms). Recall that w.r.t. to the natural addition and scalar mul- tiplication operations, Hom(V; V ) forms an F-vector space; while w.r.t.
Load more

Recommended publications
  • 1 Euclidean Vector Space and Euclidean Affi Ne Space
    Profesora: Eugenia Rosado. E.T.S. Arquitectura. Euclidean Geometry1 1 Euclidean vector space and euclidean a¢ ne space 1.1 Scalar product. Euclidean vector space. Let V be a real vector space. De…nition. A scalar product is a map (denoted by a dot ) V V R ! (~u;~v) ~u ~v 7! satisfying the following axioms: 1. commutativity ~u ~v = ~v ~u 2. distributive ~u (~v + ~w) = ~u ~v + ~u ~w 3. ( ~u) ~v = (~u ~v) 4. ~u ~u 0, for every ~u V 2 5. ~u ~u = 0 if and only if ~u = 0 De…nition. Let V be a real vector space and let be a scalar product. The pair (V; ) is said to be an euclidean vector space. Example. The map de…ned as follows V V R ! (~u;~v) ~u ~v = x1x2 + y1y2 + z1z2 7! where ~u = (x1; y1; z1), ~v = (x2; y2; z2) is a scalar product as it satis…es the …ve properties of a scalar product. This scalar product is called standard (or canonical) scalar product. The pair (V; ) where is the standard scalar product is called the standard euclidean space. 1.1.1 Norm associated to a scalar product. Let (V; ) be a real euclidean vector space. De…nition. A norm associated to the scalar product is a map de…ned as follows V kk R ! ~u ~u = p~u ~u: 7! k k Profesora: Eugenia Rosado, E.T.S. Arquitectura. Euclidean Geometry.2 1.1.2 Unitary and orthogonal vectors. Orthonormal basis. Let (V; ) be a real euclidean vector space. De…nition.
    [Show full text]
  • Glossary of Linear Algebra Terms
    INNER PRODUCT SPACES AND THE GRAM-SCHMIDT PROCESS A. HAVENS 1. The Dot Product and Orthogonality 1.1. Review of the Dot Product. We first recall the notion of the dot product, which gives us a familiar example of an inner product structure on the real vector spaces Rn. This product is connected to the Euclidean geometry of Rn, via lengths and angles measured in Rn. Later, we will introduce inner product spaces in general, and use their structure to define general notions of length and angle on other vector spaces. Definition 1.1. The dot product of real n-vectors in the Euclidean vector space Rn is the scalar product · : Rn × Rn ! R given by the rule n n ! n X X X (u; v) = uiei; viei 7! uivi : i=1 i=1 i n Here BS := (e1;:::; en) is the standard basis of R . With respect to our conventions on basis and matrix multiplication, we may also express the dot product as the matrix-vector product 2 3 v1 6 7 t î ó 6 . 7 u v = u1 : : : un 6 . 7 : 4 5 vn It is a good exercise to verify the following proposition. Proposition 1.1. Let u; v; w 2 Rn be any real n-vectors, and s; t 2 R be any scalars. The Euclidean dot product (u; v) 7! u · v satisfies the following properties. (i:) The dot product is symmetric: u · v = v · u. (ii:) The dot product is bilinear: • (su) · v = s(u · v) = u · (sv), • (u + v) · w = u · w + v · w.
    [Show full text]
  • Vector Differential Calculus
    KAMIWAAI – INTERACTIVE 3D SKETCHING WITH JAVA BASED ON Cl(4,1) CONFORMAL MODEL OF EUCLIDEAN SPACE Submitted to (Feb. 28,2003): Advances in Applied Clifford Algebras, http://redquimica.pquim.unam.mx/clifford_algebras/ Eckhard M. S. Hitzer Dept. of Mech. Engineering, Fukui Univ. Bunkyo 3-9-, 910-8507 Fukui, Japan. Email: [email protected], homepage: http://sinai.mech.fukui-u.ac.jp/ Abstract. This paper introduces the new interactive Java sketching software KamiWaAi, recently developed at the University of Fukui. Its graphical user interface enables the user without any knowledge of both mathematics or computer science, to do full three dimensional “drawings” on the screen. The resulting constructions can be reshaped interactively by dragging its points over the screen. The programming approach is new. KamiWaAi implements geometric objects like points, lines, circles, spheres, etc. directly as software objects (Java classes) of the same name. These software objects are geometric entities mathematically defined and manipulated in a conformal geometric algebra, combining the five dimensions of origin, three space and infinity. Simple geometric products in this algebra represent geometric unions, intersections, arbitrary rotations and translations, projections, distance, etc. To ease the coordinate free and matrix free implementation of this fundamental geometric product, a new algebraic three level approach is presented. Finally details about the Java classes of the new GeometricAlgebra software package and their associated methods are given. KamiWaAi is available for free internet download. Key Words: Geometric Algebra, Conformal Geometric Algebra, Geometric Calculus Software, GeometricAlgebra Java Package, Interactive 3D Software, Geometric Objects 1. Introduction The name “KamiWaAi” of this new software is the Romanized form of the expression in verse sixteen of chapter four, as found in the Japanese translation of the first Letter of the Apostle John, which is part of the New Testament, i.e.
    [Show full text]
  • 18.700 JORDAN NORMAL FORM NOTES These Are Some Supplementary Notes on How to Find the Jordan Normal Form of a Small Matrix. Firs
    18.700 JORDAN NORMAL FORM NOTES These are some supplementary notes on how to find the Jordan normal form of a small matrix. First we recall some of the facts from lecture, next we give the general algorithm for finding the Jordan normal form of a linear operator, and then we will see how this works for small matrices. 1. Facts Throughout we will work over the field C of complex numbers, but if you like you may replace this with any other algebraically closed field. Suppose that V is a C-vector space of dimension n and suppose that T : V → V is a C-linear operator. Then the characteristic polynomial of T factors into a product of linear terms, and the irreducible factorization has the form m1 m2 mr cT (X) = (X − λ1) (X − λ2) ... (X − λr) , (1) for some distinct numbers λ1, . , λr ∈ C and with each mi an integer m1 ≥ 1 such that m1 + ··· + mr = n. Recall that for each eigenvalue λi, the eigenspace Eλi is the kernel of T − λiIV . We generalized this by defining for each integer k = 1, 2,... the vector subspace k k E(X−λi) = ker(T − λiIV ) . (2) It is clear that we have inclusions 2 e Eλi = EX−λi ⊂ E(X−λi) ⊂ · · · ⊂ E(X−λi) ⊂ .... (3) k k+1 Since dim(V ) = n, it cannot happen that each dim(E(X−λi) ) < dim(E(X−λi) ), for each e e +1 k = 1, . , n. Therefore there is some least integer ei ≤ n such that E(X−λi) i = E(X−λi) i .
    [Show full text]
  • (VI.E) Jordan Normal Form
    (VI.E) Jordan Normal Form Set V = Cn and let T : V ! V be any linear transformation, with distinct eigenvalues s1,..., sm. In the last lecture we showed that V decomposes into stable eigenspaces for T : s s V = W1 ⊕ · · · ⊕ Wm = ker (T − s1I) ⊕ · · · ⊕ ker (T − smI). Let B = fB1,..., Bmg be a basis for V subordinate to this direct sum and set B = [T j ] , so that k Wk Bk [T]B = diagfB1,..., Bmg. Each Bk has only sk as eigenvalue. In the event that A = [T]eˆ is s diagonalizable, or equivalently ker (T − skI) = ker(T − skI) for all k , B is an eigenbasis and [T]B is a diagonal matrix diagf s1,..., s1 ;...; sm,..., sm g. | {z } | {z } d1=dim W1 dm=dim Wm Otherwise we must perform further surgery on the Bk ’s separately, in order to transform the blocks Bk (and so the entire matrix for T ) into the “simplest possible” form. The attentive reader will have noticed above that I have written T − skI in place of skI − T . This is a strategic move: when deal- ing with characteristic polynomials it is far more convenient to write det(lI − A) to produce a monic polynomial. On the other hand, as you’ll see now, it is better to work on the individual Wk with the nilpotent transformation T j − s I =: N . Wk k k Decomposition of the Stable Eigenspaces (Take 1). Let’s briefly omit subscripts and consider T : W ! W with one eigenvalue s , dim W = d , B a basis for W and [T]B = B.
    [Show full text]
  • Your PRINTED Name Is: Please Circle Your Recitation
    18.06 Professor Edelman Quiz 3 December 3, 2012 Grading 1 Your PRINTED name is: 2 3 4 Please circle your recitation: 1 T 9 2-132 Andrey Grinshpun 2-349 3-7578 agrinshp 2 T 10 2-132 Rosalie Belanger-Rioux 2-331 3-5029 robr 3 T 10 2-146 Andrey Grinshpun 2-349 3-7578 agrinshp 4 T 11 2-132 Rosalie Belanger-Rioux 2-331 3-5029 robr 5 T 12 2-132 Georoy Horel 2-490 3-4094 ghorel 6 T 1 2-132 Tiankai Liu 2-491 3-4091 tiankai 7 T 2 2-132 Tiankai Liu 2-491 3-4091 tiankai 1 (16 pts.) a) (4 pts.) Suppose C is n × n and positive denite. If A is n × m and M = AT CA is not positive denite, nd the smallest eigenvalue of M: (Explain briey.) Solution. The smallest eigenvalue of M is 0. The problem only asks for brief explanations, but to help students understand the material better, I will give lengthy ones. First of all, note that M T = AT CT A = AT CA = M, so M is symmetric. That implies that all the eigenvalues of M are real. (Otherwise, the question wouldn't even make sense; what would the smallest of a set of complex numbers mean?) Since we are assuming that M is not positive denite, at least one of its eigenvalues must be nonpositive. So, to solve the problem, we just have to explain why M cannot have any negative eigenvalues. The explanation is that M is positive semidenite.
    [Show full text]
  • Basics of Linear Algebra
    Basics of Linear Algebra Jos and Sophia Vectors ● Linear Algebra Definition: A list of numbers with a magnitude and a direction. ○ Magnitude: a = [4,3] |a| =sqrt(4^2+3^2)= 5 ○ Direction: angle vector points ● Computer Science Definition: A list of numbers. ○ Example: Heights = [60, 68, 72, 67] Dot Product of Vectors Formula: a · b = |a| × |b| × cos(θ) ● Definition: Multiplication of two vectors which results in a scalar value ● In the diagram: ○ |a| is the magnitude (length) of vector a ○ |b| is the magnitude of vector b ○ Θ is the angle between a and b Matrix ● Definition: ● Matrix elements: ● a)Matrix is an arrangement of numbers into rows and columns. ● b) A matrix is an m × n array of scalars from a given field F. The individual values in the matrix are called entries. ● Matrix dimensions: the number of rows and columns of the matrix, in that order. Multiplication of Matrices ● The multiplication of two matrices ● Result matrix dimensions ○ Notation: (Row, Column) ○ Columns of the 1st matrix must equal the rows of the 2nd matrix ○ Result matrix is equal to the number of (1, 2, 3) • (7, 9, 11) = 1×7 +2×9 + 3×11 rows in 1st matrix and the number of = 58 columns in the 2nd matrix ○ Ex. 3 x 4 ॱ 5 x 3 ■ Dot product does not work ○ Ex. 5 x 3 ॱ 3 x 4 ■ Dot product does work ■ Result: 5 x 4 Dot Product Application ● Application: Ray tracing program ○ Quickly create an image with lower quality ○ “Refinement rendering pass” occurs ■ Removes the jagged edges ○ Dot product used to calculate ■ Intersection between a ray and a sphere ■ Measure the length to the intersection points ● Application: Forward Propagation ○ Input matrix * weighted matrix = prediction matrix http://immersivemath.com/ila/ch03_dotprodu ct/ch03.html#fig_dp_ray_tracer Projections One important use of dot products is in projections.
    [Show full text]
  • Advanced Linear Algebra (MA251) Lecture Notes Contents
    Algebra I – Advanced Linear Algebra (MA251) Lecture Notes Derek Holt and Dmitriy Rumynin year 2009 (revised at the end) Contents 1 Review of Some Linear Algebra 3 1.1 The matrix of a linear map with respect to a fixed basis . ........ 3 1.2 Changeofbasis................................... 4 2 The Jordan Canonical Form 4 2.1 Introduction.................................... 4 2.2 TheCayley-Hamiltontheorem . ... 6 2.3 Theminimalpolynomial. 7 2.4 JordanchainsandJordanblocks . .... 9 2.5 Jordan bases and the Jordan canonical form . ....... 10 2.6 The JCF when n =2and3 ............................ 11 2.7 Thegeneralcase .................................. 14 2.8 Examples ...................................... 15 2.9 Proof of Theorem 2.9 (non-examinable) . ...... 16 2.10 Applications to difference equations . ........ 17 2.11 Functions of matrices and applications to differential equations . 19 3 Bilinear Maps and Quadratic Forms 21 3.1 Bilinearmaps:definitions . 21 3.2 Bilinearmaps:changeofbasis . 22 3.3 Quadraticforms: introduction. ...... 22 3.4 Quadraticforms: definitions. ..... 25 3.5 Change of variable under the general linear group . .......... 26 3.6 Change of variable under the orthogonal group . ........ 29 3.7 Applications of quadratic forms to geometry . ......... 33 3.7.1 Reduction of the general second degree equation . ....... 33 3.7.2 The case n =2............................... 34 3.7.3 The case n =3............................... 34 1 3.8 Unitary, hermitian and normal matrices . ....... 35 3.9 Applications to quantum mechanics . ..... 41 4 Finitely Generated Abelian Groups 44 4.1 Definitions...................................... 44 4.2 Subgroups,cosetsandquotientgroups . ....... 45 4.3 Homomorphisms and the first isomorphism theorem . ....... 48 4.4 Freeabeliangroups............................... 50 4.5 Unimodular elementary row and column operations and the Smith normal formforintegralmatrices .
    [Show full text]
  • A Guided Tour to the Plane-Based Geometric Algebra PGA
    A Guided Tour to the Plane-Based Geometric Algebra PGA Leo Dorst University of Amsterdam Version 1.15{ July 6, 2020 Planes are the primitive elements for the constructions of objects and oper- ators in Euclidean geometry. Triangulated meshes are built from them, and reflections in multiple planes are a mathematically pure way to construct Euclidean motions. A geometric algebra based on planes is therefore a natural choice to unify objects and operators for Euclidean geometry. The usual claims of `com- pleteness' of the GA approach leads us to hope that it might contain, in a single framework, all representations ever designed for Euclidean geometry - including normal vectors, directions as points at infinity, Pl¨ucker coordinates for lines, quaternions as 3D rotations around the origin, and dual quaternions for rigid body motions; and even spinors. This text provides a guided tour to this algebra of planes PGA. It indeed shows how all such computationally efficient methods are incorporated and related. We will see how the PGA elements naturally group into blocks of four coordinates in an implementation, and how this more complete under- standing of the embedding suggests some handy choices to avoid extraneous computations. In the unified PGA framework, one never switches between efficient representations for subtasks, and this obviously saves any time spent on data conversions. Relative to other treatments of PGA, this text is rather light on the mathematics. Where you see careful derivations, they involve the aspects of orientation and magnitude. These features have been neglected by authors focussing on the mathematical beauty of the projective nature of the algebra.
    [Show full text]
  • Basics of Euclidean Geometry
    This is page 162 Printer: Opaque this 6 Basics of Euclidean Geometry Rien n'est beau que le vrai. |Hermann Minkowski 6.1 Inner Products, Euclidean Spaces In a±ne geometry it is possible to deal with ratios of vectors and barycen- ters of points, but there is no way to express the notion of length of a line segment or to talk about orthogonality of vectors. A Euclidean structure allows us to deal with metric notions such as orthogonality and length (or distance). This chapter and the next two cover the bare bones of Euclidean ge- ometry. One of our main goals is to give the basic properties of the transformations that preserve the Euclidean structure, rotations and re- ections, since they play an important role in practice. As a±ne geometry is the study of properties invariant under bijective a±ne maps and projec- tive geometry is the study of properties invariant under bijective projective maps, Euclidean geometry is the study of properties invariant under certain a±ne maps called rigid motions. Rigid motions are the maps that preserve the distance between points. Such maps are, in fact, a±ne and bijective (at least in the ¯nite{dimensional case; see Lemma 7.4.3). They form a group Is(n) of a±ne maps whose corresponding linear maps form the group O(n) of orthogonal transformations. The subgroup SE(n) of Is(n) corresponds to the orientation{preserving rigid motions, and there is a corresponding 6.1. Inner Products, Euclidean Spaces 163 subgroup SO(n) of O(n), the group of rotations.
    [Show full text]
  • Math 102 -- Linear Algebra I -- Study Guide
    Math 102 Linear Algebra I Stefan Martynkiw These notes are adapted from lecture notes taught by Dr.Alan Thompson and from “Elementary Linear Algebra: 10th Edition” :Howard Anton. Picture above sourced from (http://i.imgur.com/RgmnA.gif) 1/52 Table of Contents Chapter 3 – Euclidean Vector Spaces.........................................................................................................7 3.1 – Vectors in 2-space, 3-space, and n-space......................................................................................7 Theorem 3.1.1 – Algebraic Vector Operations without components...........................................7 Theorem 3.1.2 .............................................................................................................................7 3.2 – Norm, Dot Product, and Distance................................................................................................7 Definition 1 – Norm of a Vector..................................................................................................7 Definition 2 – Distance in Rn......................................................................................................7 Dot Product.......................................................................................................................................8 Definition 3 – Dot Product...........................................................................................................8 Definition 4 – Dot Product, Component by component..............................................................8
    [Show full text]
  • Clifford Algebra with Mathematica
    Clifford Algebra with Mathematica J.L. ARAGON´ G. ARAGON-CAMARASA Universidad Nacional Aut´onoma de M´exico University of Glasgow Centro de F´ısica Aplicada School of Computing Science y Tecnolog´ıa Avanzada Sir Alwyn William Building, Apartado Postal 1-1010, 76000 Quer´etaro Glasgow, G12 8QQ Scotland MEXICO UNITED KINGDOM [email protected] [email protected] G. ARAGON-GONZ´ ALEZ´ M.A. RODRIGUEZ-ANDRADE´ Universidad Aut´onoma Metropolitana Instituto Polit´ecnico Nacional Unidad Azcapotzalco Departamento de Matem´aticas, ESFM San Pablo 180, Colonia Reynosa-Tamaulipas, UP Adolfo L´opez Mateos, 02200 D.F. M´exico Edificio 9. 07300 D.F. M´exico MEXICO MEXICO [email protected] [email protected] Abstract: The Clifford algebra of a n-dimensional Euclidean vector space provides a general language comprising vectors, complex numbers, quaternions, Grassman algebra, Pauli and Dirac matrices. In this work, we present an introduction to the main ideas of Clifford algebra, with the main goal to develop a package for Clifford algebra calculations for the computer algebra program Mathematica.∗ The Clifford algebra package is thus a powerful tool since it allows the manipulation of all Clifford mathematical objects. The package also provides a visualization tool for elements of Clifford Algebra in the 3-dimensional space. clifford.m is available from https://github.com/jlaragonvera/Geometric-Algebra Key–Words: Clifford Algebras, Geometric Algebra, Mathematica Software. 1 Introduction Mathematica, resulting in a package for doing Clif- ford algebra computations. There exists some other The importance of Clifford algebra was recognized packages and specialized programs for doing Clif- for the first time in quantum field theory.
    [Show full text]