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Gruenberg K.W., Weir A.J. — Linear Geometry
Gruenberg K.W., Weir A.J. — Linear Geometry

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Название: Linear Geometry

Авторы: Gruenberg K.W., Weir A.J.

Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: Second Edition

Год издания: 1977

Количество страниц: 212

Добавлена в каталог: 13.09.2007

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$(A,D_\varphi)$ or $(A,\bot)$, generalized similarity euclidean geometry      131 132
$(A,\varphi)$ or A, generalized affine geometry      35
$(Aut \mathscr{P}(V))_H)$, automorphisms of $\mathscr{P}(V)$ leaving H invariant      61
$(a^1,...,a^n)$, dual basis of $(a_1,...,a_n)$      80
$(f;(v_i),(v^{'}_j))$, matrix of f with respect to ordered bases $(v_i)$, $(v^{'}_j)$      72
$(f_s)_{s\in S}$, family      2
$(P,\psi)$ or P, generalized projective geometry      35
$(Pr \mathscr{P}(V))_H)$, collineations of $\mathscr{P}(V)$ leaving H invariant      61
$(Pr \mathscr{P}(V))_{H,\perp})$, collineations of $\mathscr{P}(V)$ which restrict to collineations of $\mathscr{P}(H)$ preserving $\perp$      145
$+(M_i:i\in I)$, $M_1 +...+ M_n$, sum      7
$0_V$, zero element of V      5
$Aut(V, \sigma)$, orthogonal group on euclidean space $(V, \sigma)$      144
$Aut(V,D)$, automorphisms of V preserving D      144
$Aut(W,\sigma)$, unitary group on Hilbert space $(W, \sigma)$      146
$A\zeta$, matrix $(a_{ij}\zeta)$      124
$A^t$, transpose of A      77
$A^{-1}$, inverse of A      73
$cr(A_0,A_1;A_2,A_3)$, cross-ratio      133 (ex. 1)
$diag(a_{11},...,a_{nn})$, diagonal matrix      104
$dim_F V$, dimension of V over F      10
$E_x(f)$, eigenspace of f corresponding to eigenvalue x      135
$E_{ij}$, matrix with (i,j)-term 1 and others 0      74 (ex. 2)
$f:S\rightarrow T$, mapping f of S into T      2
$F^n$, n-tuples      5
$f^t$, transpose of f      83 (ex. 7)
$f^{-1}$, the inverse of f      3 (ex. 1)
$F^{m\times n}$, $m\times n$ matrices      72
$J(S_i:i\in I)$, $S_1 J S_2$, join      16
$M^\circ$, annihilator of M      81
$M^{\bot(\sigma)}$, $M^{\top(\sigma)}$, subspaces orthogonal to M      93
$m_a(X)$, minimum polynomial at a      151
$Q(\sigma)$, quadric      111
$S = T$, S equals T      2
$S\subset T$, S is contained in T      1
$t_a$, translation $v \rightarrow v + a$      42
$U_{\langle C \rangle}$, complexification of real space U      14
$V^*$, dual space of V      80
$V_f$, F[X]-module defined by $f:V\rightarrow V$      151
$x\in S$, x is an element of S      1
$[S]_f$, submodule of $V_f$ generated by S      153
$\cap(M_i:i\in I)$, $M_1\cap...\cap M_n$, intersection      3
$\cup(M_i:i\in I)$, $M_1\cup...\cup M_n$, union      3
$\delta_{ij}$, Kronecker delta      73
$\imath$, canonical isomorphism $V \rightarrow V^{**}$      81
$\mathbb{C}$, complex numbers      1
$\mathbb{F}_p$, integers modulo p      1
$\mathbb{Q}$, rational numbers      1
$\mathbb{R}$, real numbers      1
$\mathbb{Z}$, integers      1
$\mathfrak{B}^*$, dual proposition      84
$\mathfrak{C}^*$, dual configuration      84
$\mathscr{A}(f)$, mapping of affine geometries induced by $f:S\rightarrow S^{'}$      42
$\mathscr{A}(S)$, affine geometry on coset S      16
$\mathscr{A}(S,d)$, euclidean geometry on coset S with distance d      127
$\mathscr{A}(S,D)$, similarity euclidean geometry on coset S with distance class D      131
$\mathscr{A}(t_{-a}gt_{a^{'}})$, affinity      42
$\mathscr{B}(V)$, bilinear forms on V      89
$\mathscr{B}^+(V)$, symmetric bilinear forms on V      101 (ex. 4)
$\mathscr{B}^-(V)$, skew-symmetric bilinear forms on V      101 (ex. 4)
$\mathscr{L}_F(V,V^{'})$, linear mappings of V into $V^{'}$      68
$\mathscr{P}(f)$, mapping of projective geometries induced by $f:V\rightarrow V^{'}$      45
$\mathscr{P}(V)$, projective geometry on vector space V      29
$\mathscr{P}^*(V)$, dual geometry      86
$\mathscr{Q}(V)$, quadratic forms on V      102 (ex. 9)
$\neq$, is not equal to      2
$\not\in$, is not an element of      2
$\not\subset$, is not contained in      2
$\oplus(M_i:i\in I)$, $M_1\oplus...\oplus M_n$, direct sum      7
$\sigma_M$, restriction of $\sigma$ to subspace M      95
$\underline{\sigma}$, $\tilde{\sigma}$, linear mappings induced by the bilinear form $\sigma$      91
$\widehat{BAC}$, angle      128
$\{e_1,...,e_n\}$, standard basis of $F^n$      13
$\{E_{ij}\}$, standard basis of $F^{m\times n}$      74 (ex. 2)
$\|a\|$, norm of a      125
(d), class of distances similar to d      131
0, zero matrix      72
a(S), annihilator (ideal) of S      151
Af A, affine group on A      59
Affine geometry      see "Geometry"
Affinity      42ff
Affinity, degenerate      70ff
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 (ex. 3)
Associative law for matrix multiplication      73
Aut V, general linear group      59
Aut X, automorphisms on X; X = V, F, A or P      59
Automorphism groups for V, A, P, F      59
Automorphism of euclidean geometry      140ff
Automorphism of field      56 58 2) 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
Axis, of central collineation      48 (exx. 3—5)
Basis (base)      9ff
Basis cartesian      127 142 146
Basis dual      80
Basis of torsion-free F[X]-module      163
Basis standard, of $F[X]^n$      168
Basis standard, of $F^n$      12 (ex. 9) 13
Basis standard, of $F^{m\timesn}$      74 (ex. 2)
Bilinear form      89ff
Bilinear form, alternating      99
Bilinear form, degenerate      91 92
Bilinear form, negative definite      106
Bilinear form, orthosymmetric      97 101
Bilinear form, positive definite      10(1
Bilinear form, skew-symmetric      99 101 4)
Bilinear form, symmetric      97 101 4 8 9)
Bilinear polynomial      90
C(Q), representative cone of projective quadric Q      115
Canonical matrix rational      166
Canonical matrix, Jordan (classical)      167
Cartesian basis      127 142 146
Center of (central) collineation      48 (exx. 3—6) 54 63 83
Center of affine polarity      111 (ex. 5)
Center of affine quadric      119
Center of perspective of two triangles      23
Center of perspectivity      48 (ex. 6) 65
Center of sphere      133 (ex. 3)
Central collineation      48 (exx. 3—6) 54 62ff 83
Central collineation, center of      see "Center"
Central collineation, hyperplane of      63
Centroid      48 (ex. 2)
Characteristic      2
Characteristic (Fano's Theorem)      40
Characteristic, field of      40 99ff
Classical (Jordan) canonical matrix      167
Classification of collineations      170ff
Coefficient space      50
Collineation      45 170ff
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
Cone, representative      115
Congruence of bilinear forms      89 90 103ff
Congruence of matrices      86 90 103ff
conic      111 114 120 2)
Conic section      116
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)      15ff
Coset parallel      18
Coset, dimension of      16
Coset, space of      67 154
Coset, subspace belonging to      16
Cross-ratio      132 133 2 5 6)
d(a,b), distance from a to b      125
Degenerate, affine polarity      111 (ex. 6)
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 (ex. 1) 95
Desargue's Theorem (affine)      23
Desargue's Theorem (projective)      32 (ex. 6) 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 (ex. 3) 49 54 63 75
Dimension of a coset      16
Dimension of a projective geometry      30
Dimension of a vector space      10ff
Dimension of an affine geometry      16
Dimension, finite      9ff
Dimension, projective      30
Direct sum of submodules      155ff
Direct sum of subspaces      7 8 8) 11 12 95 96 102ff 155ff
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 11) 75
Dual basis      80
Dual configuration      84
Dual frame of reference      87 88
Dual geometry      86ff
Dual space      80
Dual, proposition      84
Dual, self-      171 172
Duality      85
Duality, principle of      84
Eigenspace      71 (ex. 4) 135 142 143
Eigenvalue      71 (ex. 4) 135 137 141ff
Eigenvector      71 (ex. 4) 135ff 140 141ff
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 2)
Elementary transformation      70 (ex. 10)
Embedding theorem      32 52 61 116 130
Equation linear      49ff 79ff 82
Equation of affine quadric      117ff
Equation of euclidean quadric      138ff
Equation of projective quadric      112ff
Equivalence of matrices      77 165
Exchange Lemma      9
Exponent of commutative group      153 (ex. 4)
Family      2
Fano's Theorem      40
Field      4 5
Field of characteristic      2 40 99ff 115)
Field of gaussian numbers      14 (ex. 1)
Field of quotients      160
Field, ground      12
Finite dimensional vector space      9ff
Finite field      109 (exx. 8—11) 162 174
Finite geometry (affine)      20 (ex. 1) 21) 38
Finite geometry (projective)      32 (exx. 4 5) 40 41 119 174
Form Hermitian      122ff
Form linear      80ff 101
Form quadratic      100ff
Form sesquilinear      121ff
Frame of reference for $\mathscr{P}(V)$ 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 $\mathscr{A}(F^n)$      43
Frame of reference, standard, for $\mathscr{P}(F^{n+1})$      47
Function      see "Mapping"
F[X], polynomials in X      6
Gaussian number field      14 (ex. 1)
Generator of a projective quadric      114 (exx. 11—14)
Geometry, "affine, of dimension n over F"      22
Geometry, "projective, of dimension n over F"      32
Geometry, affine      16ff 35
Geometry, affine, in $\mathscr{P}(V)$ determined by H      36 61 130
Geometry, Euclidean      127ff
Geometry, projective      29ff 35
Geometry, similarity euclidean      13Iff
Geometry, similarity euclidean, determined in $\mathscr{P}(V)$ by H and D      132
Greatest common divisor      161 (exx. 3 6)
Ground field      12ff
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 range, (affine)      27
Harmonic range, (projective)      40 113 133
Hermitian form      122ff 124 3)
Hermitian mapping      147 (ex. 6)
Hermitian matrix      124 (ex. 2)
Hilbert space      146ff
Homogeneous (projective) coordinate row      47
Homogeneous linear equations      50
Homogeneous vector      22ff 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      63ff
I, identity matrix      73
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
1 2
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