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Baer R. — Linear Algebra and Projective Geometry

Baer R. — Linear Algebra and Projective Geometry

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Название: Linear Algebra and Projective Geometry

Автор: Baer R.

Аннотация:

PURE AND APPLIED MATHEMATICS
A Series of Monographs and Textbooks
Edited by
Paul A. Smith and Samuel Eilenberg Columbia University, New York

In this book we intend to establish the essential structural identity of projective geometry and linear algebra. It has, of course, long been realized that these two disciplines are identical. The evidence substantiating this statement is contained in a number of theorems showing that certain geometrical concepts may be represented in algebraic fashion. However, it is rather difficult to locate these fundamental existence theorems in the literature in spite of their importance and great usefulness. The core of our discussion will consequently be formed by theorems of just this type. These are concerned with the representation of projective geometries by linear manifolds, of projectivities by semi-linear transformations, of collineations by linear transformations and of dualities by semi-bilinear forms. These theorems will lead us to a reconstruction of the geometry which was the starting point of our discourse within such (apparently) purely algebraic structures as the endomorphism ring of the underlying linear manifold or the full linear group.

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Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1952

Количество страниц: 322

Добавлена в каталог: 07.05.2010

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Предметный указатель

Absorption law      312
Adjoint space      25—26
Adjunct space      36
Admissible, involutions of first and second kind      147
Admissible, projectivities      144
Annulet      172
Anti-automorphism      78
Anti-homomorphism      103
Anti-isomorphism      96
Anti-semi-linear transformation      99 190
Auto-projectivities, group of      52
Basis      13 265
Betweenness      223
Bilinear form      101
Canonical dual space      98
Center of a group      201
Center of a perspectivity      292
Centralizer, first      63 203
Centralizer, second      203
Characteristic two, criteria for Fano’s postulate      38 IV. 114 IV. 116
Class of a linear transformation      207
Collineation      62
Commutativity, criteria for, of the field of scalars      III. 3. Proposition 2 67 Second 68 Postulate 71 III. 75—76 III. 87 IV. 106 IV. 108 IV. 114 IV. 132 IV. 141 IV. 143
Complement      12
Complementation principle      260
Complementation theorem      12
Conjugate numbers in fields      72
Containedness      8
Cross ratio      71—72
Crossed homomorphisms      248
Decomposition endomorphisms      169
Dedekind’s (or modular) law      9 259
Dependence of elements (vectors)      13
Dependence of points      41 264
Desargues’ theorem      266—267
Dilatation      53 57
DIMENSION      16 265
Direct sum      12
Dualities      32 95 310
Duality principle of projective geometry      100 263
Duality theorem      97
Euclidean fields      144
Euclidean fields, anti-isomorphisms of endomorpliism rings      192—193
Existence theorem for basis      14
Existence theorem for dualities      96
Existence theorem for polarities      159
Fano’s postulate      37
Field      7
Finiteness, criteria for, of rank      II. 3. Corollary 1 26 II. 28 II. 30 IV. 96 IV. 97 IV. 97 IV. 103 IV. 108 IV. 111 V. 182 V. 189 V. 192 V. 193 VI. 243 VI. 250 VI. 252
Flock      3 303
Formally real fields      144
Forms, bilinear      101
Forms, linear      25
Forms, semi-bilinear      101
Function space [F,C]      19
Fundamental theorem of projective geometry, complement to second      III. 4. Theorem 2 87
Fundamental theorem of projective geometry, First      44
Fundamental theorem of projective geometry, Second      68
Galois correspondence between a space and its adjoint space      28
Galois correspondence between a space and its group      VI. 4. Theorem 4 227
Galois correspondence between a space and its ring      172—173
Galois correspondence between direct decomposition into points and maximal groups of involutions      VI. App. I. Remark 1 244
Geometry defined by a bilinear form      IV. 5. Remark 3 152
Group automorphism of endomorphism ring      V. 4. Theorem 2 187
Group automorphism of full linear group      237
Group automorphism of full semi-linear group      256
Group homothetic      52
Group of a duality      144
Group of a hyperplane      292
Group of a linear manifold (survey)      63
Group of collineations      62
Group of involutions      237
Group of multiplications      52 63 246
Group, extended, of automorphisms of endomorphism ring      196
Harmonic set      III. 4. Remark 5 76
Homothetic transformation      53 57
Hyperplane      19 264
Ideals      172
Idempotents      169
Imbedding theorem      268
inclusion      8
Independent elements (vectors)      13
Independent points      41 264

Index of inertia (Sylvester)      130
Index of polarity      121
Induced isomorphisms of first kind of full linear group      229
Induced isomorphisms of first kind of full semi-linear group      249
Induced isomorphisms of second kind of full linear group      229
Induced isomorphisms of second kind of full semi-linear group      251
Intersection of subspaces      9
Involutions      144
Involutorial mappings      111
Isomorphism law      11
Isomorphism theorem for endomorphism ring      V. 4. Theorem 1 183
Isomorphism theorem for full linear group      231
Isomorphism theorem for full semi-linear group      252
Isotropic subspaces      113
Isotropic subspaces, strictly      109 113
Kernel      168
Line      18 71 265
Linear anti-form      36
Linear dependence and independence      13
Linear form      25
Linear manifold      7
Linear mapping      91
Linear transformation      42
Maximal elements in partially ordered sets      310
Maximum principle of set theory      310
Multiplications, characterization of      III. 1. Proposition 3 43
Multiplications, group of      52 63 246
Non-lsotropic subspaces      101 113
normalizer      219
Null systems and N-sub spaces      106
Nullity of a polarity      116
Nullity of an endomorphism      168
Ordering, algebraic, of a field      94 127
Ordering, projective, of a space      94
Orthogonal group      158
Orthogonal idempotents      169
Paired spaces      35
Pappus, postulate of      71
Pappus, property of      69
Partially ordered set      258 309
Pascal, theorem of      143
Pascalean polarity      139
permutations      309
Perspectivity      64 292
Plane      18 265
Point      18 260
Polarity, pole-polar relation      109
Positivity, domain of      93 127
Preservation of forms by transformations      151
Projection of aline from a point      91
Projectivity      40 310
Pythagoras, Theorem of      152
Quotient spaces      10
Rank formulas for a pair of subspaces      17—18
Rank formulas for a subspace and its dual      33
Rank of a subspace      16 265
Rank of an endomorphism      168
Regular rings      179
Ring of endomorphisms of a linear manifold      169
Self-duality theorem      97
Semi-automorphism      83
Semi-bilinear form      101
Semi-linear transformation      42
simplex      66
Singular automorphisms of full linear group      230
Singular automorphisms of full semi-linear group      247
Strictly isotropic      113
Structure theorem for endomorphism ring of linear manifolds      183
Structure theorem for full linear groups      229
Structure theorem for full semi-linear groups      247
Structure theorem for linear manifolds      16
Structure theorem for projective spaces      51
subspace      8
Sum of subspaces      9 259
Surveyof groups of a linear manifold      63
Sylvester’s theorem of inertia      129
Symmetrical forms      111
Symplectic group      155
Transitive groups      153
Translation      53 57
Uniqueness theorem for anti-semi-linear transformations      194
Uniqueness theorem for polarities      148 159
Uniqueness theorem for rank      14
Unitary group      158

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