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| Gruenberg K.W. — Linear Geometry |
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| Предметный указатель |
Affine geometry see “Geometry”
Affinity 42
Affinity, degenerate 70
angle 126 128
Annihilator 81
Annihilator, ideal 151
Annihilator, mapping 81 94
Anti-automorphism 120 122
Anti-isomorphism 84
Associative law for mapping multiplication 3
Associative law for matrix multiplication 73
Automorphism of euclidean geometry 14
Automorphism of field 56 58 65 123
Automorphism of Hilbert space 146
Automorphism of R-module 154
Automorphism of sesquilinear space 124
Automorphism of similarity euclidean geometry 144
Automorphism of vector space 66
Automorphism, groups for  , A, P, F 59
Axis of central collineation 48
Basis (base) 9
Basis (base) of torsion-free ![$\mathrm{F}[\mathrm{X}]$](/math_tex/d9bb8757cbd55600996dded96af7582082.gif) -module 163
Basis (base), cartesian 127 142 146
Basis (base), dual 80
Basis (base), standard of ![$\mathrm{F}[\mathrm{X}]^n$](/math_tex/c81b15aab43878fb5e180587e58e509a82.gif) 168
Basis (base), standard of  12 13
Basis (base), standard of  74
Bilinear form 89
Bilinear form, alternating 99
Bilinear form, degenerate 91 92
Bilinear form, negative definite 106
Bilinear form, orthosymmetric 97 101
Bilinear form, positive definite 106
Bilinear form, skew-symmetric 99 101
Bilinear form, symmetric 97 101
Bilinear polynomial 90
Canonical matrix, Jordan (classical) 167
Canonical matrix, rational 166
Cartesian basis 127 142 146
Center of (central) collineation 48 54 63 83
Center of affine polarity 111
Center of affine quadric 119
Center of perspective of two triangles 23
Center of perspectivity 48 65
Center of sphere 133
Central collineation 48 54 62 83
Central collineation, center of see “Center”
Central collineation, hyperplane of 63
Centroid 48
Characteristic 2
Characteristic (Fano's Theorem) 40 111 115
Characteristic, field of 40 99
Classical (Jordan) canonical matrix 167
Classification of collineations 170
Coefficient space 50
Collineation 45 170
Collineation, central see “Central”
Completeness of Hilbert space 147
Completion of affine quadric (to projective quadric) 117 118 120
Complexification of real bilinear form 124 146
Complexification of real vector space 14 58 112 134 136 140
Congruence of bilinear forms 89 90 103
Congruence of matrices 86 90 103
conic 111 114 120
Conic, action 116
Cono, representative 115
Coordinate row (homogeneous) of a projective point 47
Coordinate row of a vector 13
Coordinate row of an affine point 44
Coordinate system (homogeneous) for a projective geometry 46
Coordinate system for a euclidean geometry 129
Coordinate system for a vector space 13
Coordinate system for an affine geometry 44
Correlation 85 110 120
Correlation, degenerate 110
Coset (translated subspace) 15
Coset (translated subspace), dimension of 16
Coset (translated subspace), parallel 18
Coset (translated subspace), space of 67 154
Coset (translated subspace), subspace belonging to 16
Cross-ratio 132 133
Degenerate, affine polarity 111
Degenerate, affine quadric 119
Degenerate, affinity 70
Degenerate, bilinear form 91 92
Degenerate, correlation 110
Degenerate, null-polarity 110
Degenerate, polarity 110
Degenerate, projective quadric 112
Degenerate, projectivity 70
Degenerate, subspace (with respect to a bilinear form) 92 95
Desargue's Theorem (affine) 23
Desargue's Theorem (projective) 32 38 41 48 51 86
Desargue's Theorem, converse of 38
Desargue's Theorem, related to a division ring 39 40 48
Determinant 144
Dilatation 44 49 54 63 75
Dimension of a coset 16
Dimension of a projective geometry 30
Dimension of a vector space 10
Dimension of an affine geometry 16
Dimension, finite 9 15
Dimension, projective 30
Direct sum of submodules 155
Direct sum of subspaces 7 8 11 12 95 96 102 155
Distance on a coset 127
Distance on a euclidean space 125
Distance on a Hilbert space 146
Distance on a vector space 125
Distance, similar 131
Division ring 39 49 75
Dual basis 80
Dual configuration 84
Dual frame of reference 87 88
Dual geometry 86
Dual proposition 84
Dual space 80
Dual, self 171 172
Duality 85
Duality, principle of 84
Eigenspace 71 135 142 143
Eigenvalue 71 135 137 14
Eigenvector 71 135 140
Elementary divisors of a finitely generated F[X]-module 159 160
Elementary divisors of a linear mapping 166
Elementary divisors of a matrix 168 169
Elementary move 164 165
Elementary transformation 70
Embedding theorem 32 52 61 116 130
Equation of affine quadric 117
Equation of euclidean quadric 138
Equation of projective quadric 112
Equation, linear 49 79 82
Equivalence of matrices 77 165
Exchange Lemma 9
Exponent of commutative group 153
Family 2
Fano'a Theorem 40
Field 4 5
Field of characteristic 2 40 99 111 115
Field of gaussian numbers 14
Field of quotients 160
Field, ground 12
Finite dimensional vector space 9 15
Finite field 109 162 174
Finite geometry (affine) 20 29 38
Finite geometry (projective) 32 40 41 119 174
Form, Hermitian 122
Form, linear 80 101
Form, quadratic 100
| Form, sesquilinear 121
Frame of reference for  determined by a basis of V 46
Frame of reference for a euclidian geometry 129
Frame of reference for a projective geometry 46
Frame of reference for an affine geometry 43
Frame of reference standard for  47
Frame of reference standard for  43
Function see “Mapping”
Gaussian number field 14
Generator of a projective quadric 114
Geometry, affine 35
Geometry, affine (generalized) 16
Geometry, affine, in determined by H 36 61 130
Geometry, determined in by H and D 132
Geometry, Euclidean 127
Geometry, projective 35
Geometry, projective (generalized) 29
Geometry, similarity euclidean 131
Geometry, “affine of dimension n over F” 22
Geometry, “projective of dimension n over F” 32
Greatest common divisor 161
Ground field 12
Group 4
Group, abelian (commutative) 4 150 160 163
Group, affine 59 75
Group, automorphism 59
Group, collineation (projective) 59
Group, commutative see “Group abelian”
Group, general linear 59
Group, homomorphism of 59
Group, isomorphism of 59
Group, orthogonal 144
Group, permutation 4 62
Group, projective see “Group collineation”
Group, rotation 4 144
Group, unitary 146
Harmonic, conjugate, (affine) 27
Harmonic, conjugate, (projective) 40
Harmonic, Constructive, (affine) 27
Harmonic, Constructive, (projective) 40 52
Harmonic, rango, (affine) 27
Harmonic, rango, (projective) 40 113 133
Hermitian form 122 124
Hermitian mapping 147
Hermitian matrix 124
hilbert 146
Hilbert space 146
Homogeneous (projective) coordinate row 47
Homogeneous (projective) linear equations 50
Homogeneous (projective) vector 22 30
Homomorphism of groups 59
Homomorphism of R-modules 154
Homomorphism of vector spaces 66
Hyperplane at infinity 36
Hyperplane in affine geometry 16
Hyperplane in projective geometry 30
Hyperplane of a central collineation 63
Ideal 152
Ideal, annihilator 151
Ideal, principal 152
Identity element of group 4
Identity element of ring 149
Identity mapping 2 3
Identity matrix 73
Image 2
Incidence propositions (affine) 19
Incidence propositions (projective) 31
Indecomposable submodule 159
Index notation 2
Infinite dimensional vector space 9 11 68 69 82 92 130 147
Integral domain 155 160
Intersection of cosets 16
Intersection of sets 3
Intersection of subspaces 7
Invariant configuration 170
Invariant factors of linear mapping 166
Invariant factors of matrix 168 169
Invariant factors of module 156 160 165
Invariant subspace 153
Inverse of a group element 4
Inverse of a mapping 3
Inverse of a matrix 73 74
Isomorphism of affine geometries 20 52
Isomorphism of bilinear spaces 89
Isomorphism of euclidean geometries 128 130
Isomorphism of fields 56
Isomorphism of groups 59
Isomorphism of modules 154
Isomorphism of projective geometries 31 52
Isomorphism of similarity euclidean geometries 131
Isomorphism of vector spaces 12
Isomorphism, anti-isomorphism 84
Isomorphism, linear isomorphism 66 91
Isomorphism, semi-linear isomorphism 56
Join (sum) of subspaces 7
Join of cosets 16
Jordan (classical) canonical matrix 167
Jordan canonical 107
Kernel of group homomorphism 60 62
Kernel of linear mapping 66
Kernel of module homomorphism 154
Kronecker delta 69 73
Length (norm) in euclidean space 125
Length (norm) in Hilbert spaee 146
Length of element in a principal ideal domain 161
Line at infinity 36 37
Line in affine geometry 16
Line, projective 30
Linear algebra 08 136
Linear automorphism 66
linear combination 6
Linear complex 110
Linear dependence 8
Linear equations 49 79 82
Linear form 80
Linear independence, over F 9
Linear independence, over F[X] 158
Linear isomorphism 12 66 91
Linear mapping 66
Linear mapping,  -symmetric 135 140
Linear mapping, commuting 140 147
Linear mapping, similar 154
Linear mapping, trace of 75 83
Linear mapping, transpose of 83 93 169
Linear mapping, triangulable 140 147
Mapping (function) 2
Mapping (function) annihilator 81 94
Mapping (function), hermitian 147
Mapping (function), inverse 3
Mapping (function), normal 147
Mapping (function), one-one 2
Mapping (function), product of 2
Mapping (function), semi-linear 120
Mapping (function), unitary 147
Matrix 72
Matrix of bilinear form 90
Matrix of linear mapping 72
Matrix of quadratic form 102
Matrix, addition of 72
Matrix, alternating 103
Matrix, congruence of 86 103
Matrix, diagonal 104
Matrix, equivalence of 77 165
Matrix, hermitian 124
Matrix, identity 73
Matrix, inverse of 73 74
Matrix, invertible (non-singular) 74
Matrix, multiplication of 72
Matrix, nilpotent 74
Matrix, non-singular (invertible) 74
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