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Introduction to Vector Spaces, Vector Algebras, and Vector Geometries

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Introduction to Vector Spaces, Vector Algebras, and Vector Geometries Introduction to Vector Spaces, Vector Algebras, and Vector Geometries Richard A. Smith October 14, 2011 Abstract An introductory overview of vector spaces, algebras, and linear geometries over an arbitrary commutative field is given. Quotient spaces are emphasized and used in constructing the exterior and the symmetric algebras of a vector space. The exterior algebra of a vector space and that of its dual are used in treating linear geometry. Scalar product spaces, orthogonality, and the Hodge star based on a general basis are treated. Contents 1 Fundamentals of Structure 5 1.1 WhatisaVectorSpace? ..................... 5 1.2 SomeVectorSpaceExamples . 6 1.3 Subspace, Linear Combination and Span . 6 arXiv:1110.3350v1 [math.HO] 14 Oct 2011 1.4 Independent Set, Basis and Dimension . 7 1.5 SumandDirectSumofSubspaces. 12 1.6 Problems.............................. 16 2 Fundamentals of Maps 18 2.1 StructurePreservationandIsomorphism . 18 2.2 Kernel,LevelSetsandQuotientSpace . 20 2.3 ShortExactSequences . 24 2.4 Projections and Reflections on V = W ⊕ X ........... 26 2.5 Problems.............................. 28 1 3 More on Maps and Structure 30 3.1 FunctionSpacesandMapSpaces . 30 3.2 DualofaVectorSpace,DualofaBasis. 30 3.3 Annihilators............................ 32 3.4 DualofaMap........................... 33 3.5 TheContragredient. .. .. 34 3.6 ProductSpaces .......................... 34 3.7 MapsInvolvingProducts. 35 3.8 Problems.............................. 38 4 Multilinearity and Tensoring 40 4.1 Multilinear Transformations . 40 4.2 FunctionalProductSpaces . 41 4.3 FunctionalTensorProducts . 42 4.4 TensorProductsinGeneral . 43 4.5 Problems.............................. 49 5 Vector Algebras 50 5.1 The New Element: Vector Multiplication . 50 5.2 QuotientAlgebras......................... 51 5.3 AlgebraTensorProduct . 52 5.4 TheTensorAlgebrasofaVectorSpace . 53 5.5 TheExteriorAlgebraofaVectorSpace. 54 5.6 The Symmetric Algebra of a Vector Space . 61 5.7 Null Pairs and Cancellation . 63 5.8 Problems.............................. 63 6 Vector Affine Geometry 65 6.1 BasingaGeometryonaVectorSpace. 65 6.2 AffineFlats ............................ 65 6.3 Parallelism in Affine Structures . 66 6.4 TranslatesandOtherExpressions . 67 6.5 AffineSpanandAffineSum . .. .. 68 6.6 Affine Independence and Affine Frames . 69 6.7 AffineMaps ............................ 70 6.8 SomeAffineSelf-Maps ...................... 73 6.9 CongruenceUndertheAffineGroup. 74 6.10Problems.............................. 74 2 7 Basic Affine Results and Methods 75 7.1 Notations ............................. 75 7.2 Two Axiomatic Propositions . 75 7.3 ABasicConfigurationTheorem . 76 7.4 Barycenters ............................ 77 7.5 ASecondBasicConfigurationTheorem . 78 7.6 VectorRatios ........................... 81 7.7 A Vector Ratio Product and Collinearity . 83 7.8 AVectorRatioProductandConcurrency . 84 8 An Enhanced Affine Environment 86 8.1 InflatingaFlat .......................... 86 8.2 DesarguesRevisited. .. .. 89 8.3 Representing (Generalized) Subflats . 91 8.4 Homogeneous and Pl¨ucker Coordinates . 92 8.5 Dual Description Via Annihilators . 94 8.6 DirectandDualPl¨uckerCoordinates . 95 8.7 ADualExteriorProduct. .100 8.8 ResultsConcerningZeroCoordinates . 100 8.9 SomeExamplesinSpace . .102 8.10 FactoringaBlade. .. .. .105 8.11 AlgorithmforSumorIntersection . 107 8.12 Adapted Coordinates and ∨ Products. .108 8.13Problems..............................114 9 Vector Projective Geometry 115 9.1 The Projective Structure on V ..................115 9.2 ProjectiveFrames. .116 9.3 PappusRevisited .........................117 9.4 ProjectiveTransformations. .120 9.5 ProjectiveMaps..........................122 9.6 CentralProjection . .. .. .122 9.7 Problems..............................124 10 Scalar Product Spaces 125 10.1 PairingsandTheirIncludedMaps. 125 10.2 Nondegeneracy . .. .. .125 10.3 Orthogonality,ScalarProducts . 126 3 10.4 ReciprocalBasis. .127 10.5 RelatedScalarProductforDualSpace . 129 10.6 SomeNotationalConventions . 130 10.7 StandardScalarProducts . .131 10.8 OrthogonalSubspaces andHodgeStar . 132 10.9 ScalarProductonExteriorPowers . 134 10.10Another Way to Define ∗ .....................137 10.11HodgeStarsfortheDualSpace . 140 10.12Problems..............................141 4 1 Fundamentals of Structure 1.1 What is a Vector Space? A vector space is a structured set of elements called vectors. The structure required for this set of vectors is that of an Abelian group with operators from a specified field. (For us, multiplication in a field is commutative, and a number of our results will depend on this in an essential way.) If the specified field is F, we say that the vector space is over F. It is standard for the Abelian group to be written additively, and for its identity element to be denoted by 0. The symbol 0 is also used to denote the field’s zero element, but using the same symbol for both things will not be confusing. The field elements will operate on the vectors by multiplication on the left, the result always being a vector. The field elements will be called scalars and be said to scale the vectors that they multiply. The scalars are required to scale the vectors in a harmonious fashion, by which is meant that the following three rules always hold for any vectors and scalars. (Here a and b denote scalars, and v and w denote vectors.) 1. Multiplication by the field’s unity element is the identity operation: 1 · v = v. 2. Multiplication is associative: a · (b · v)=(ab) · v. 3. Multiplication distributes over addition: (a + b) · v = a · v + b · v and a · (v + w)= a · v + a · w. Four things which one might reasonably expect to be true are true. Proposition 1 If a is a scalar and v a vector, a · 0=0, 0 · v =0, (−a) · v = −(a · v), and given that a · v =0, if a =06 then v =0. Proof: a · 0= a · (0+0) = a · 0+ a · 0; adding − (a · 0) to both ends yields the first result. 0 · v = (0+0) · v =0 · v +0 · v; adding − (0 · v) to both ends yields the second result. Also, 0 = 0 · v = (a +(−a)) · v = a · v +(−a) · v; adding − (a · v) on the front of both ends yields the third result. If a · v =0 with a =6 0, then a−1 · (a · v)= a−1 · 0, so that (a−1a) · v = 0 and hence v =0 as claimed. Exercise 1.1 Given that a · v =0, if v =06 then a =0. 5 Exercise 1.2 −(a · v)= a · (−v). Because of the harmonious fashion in which the scalars operate on the vectors, the elementary arithmetic of vectors contains no surprises. 1.2 Some Vector Space Examples There are many familiar spaces that are vector spaces, although in their usual appearances they may have additional structure as well. A familiar vector space over the real numbers R is Rn, the space of n-tuples of real numbers with component-wise addition and scalar multiplication by elements of R. Closely related is another vector space over R, the space ∆Rn of all translations of Rn, the general element of which is the function from Rn to n itself that sends the general element (ξ1, ..., ξn) of R to (ξ1 + δ1, ..., ξn + δn) n for a fixed real n-tuple (δ1, ..., δn). Thus an element of ∆R uniformly vectors the elements of Rn to new positions, displacing each element of Rn by the same vector, and in this sense ∆Rn is truly a space of vectors. In the same fashion as with Rn over R, over any field F, the space F n of all n-tuples of elements of F constitutes a vector space. The space Cn of all n-tuples of complex numbers in addition to being considered to be a vector space over C may also be considered to be a (different, because it is taken over a different field) vector space over R. Another familiar algebraic structure that is a vector space, but which also has additional structure, is the space of all polynomials with coefficients in the field F. To conclude this initial collection of examples, the set consisting of the single vector 0 may be considered to be a vector space over any field; these are the trivial vector spaces. 1.3 Subspace, Linear Combination and Span Given a vector space V over the field F, U is a subspace of V, denoted U ⊳ V, if it is a subgroup of V that is itself a vector space over F. To show that a subset U of a vector space is a subspace, it suffices to show that U is closed under sum and under product by any scalar. We call V and {0} improper subspaces of the vector space V, and we call all other subspaces proper. Exercise 1.3 As a vector space over itself, R has no proper subspace. The set of all integers is a subgroup, but not a subspace. 6 Exercise 1.4 The intersection of any number of subspaces is a subspace. A linear combination of a finite set S of vectors is any sum s∈S cs · s obtained by adding together exactly one multiple, by some scalar coefficient, of each vector of S. A linear combination of an infinite set S Pof vectors is a linear combination of any finite subset of S. The empty sum which results in forming the linear combination of the empty set is taken to be 0, by convention. A subspace must contain the value of each of its linear combinations. If S is any subset of a vector space, by the span of S, denoted hSi, is meant the set of the values of all linear combinations of S. hSi is a subspace. If U = hSi, then we say that S spans U and that it is a spanning set for U. Exercise 1.5 For any set S of vectors, hSi is the intersection of all sub- spaces that contain S, and therefore hSi itself is the only subspace of hSi that contains S. 1.4 Independent Set, Basis and Dimension Within a given vector space, a dependency is said to exist in a given set of vectors if the zero vector is the value of a nontrivial (scalar coefficients not all zero) linear combination of some finite nonempty subset of the given set. A set in which a dependency exists is (linearly) dependent and otherwise is (linearly) independent.
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