线性几何
2009-10
世界图书出版公司
(德)格伦伯格
198
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This is essentially a book on linear algebra．But the approach is somewhat unusual in that we emphasise throughout the geometric aspect of the subject．The material is suitable for a course on linear algebra for mathematics majors at North American Universities in their junior or senior year and at British Universities in their second or third year．However， in view of the structure of undergraduatc courses in the United States，it is very possible that，at many institutions，the text may be found more suitable at the beginning graduate level． The book has two aims：to provide a basic course in linear algebra up to．and including，modules over a principal ideal domain；and to explain in rigorous language the intuitively familiar concepts of euclidean，affine， and projective geometry and the relations between them．It is increasingly recognised that linear algebra should be approached from a geometric point of vlew．This applies not only to mathematics majors but also to mathematically-oriented natural scientists and engineers． The material in this book has been taught for many years at Queen Mary College in the University of London and one of us has used portions of it at the University of Michigan and at Cornell University．It can be covered adequately in a ful！one．year course．But suitable parts can also be used for one．semester courses with either a geometric or a purely algebraic flavor. We shall give below explicit and detailed suggestions on how this can be done（in the“Guide to the Reader”)． The first chapter contains in fairly concise form the definition and most elementary properties of a vector space．Chapter 2 then defines affine and projective geometries in terms of vector spaces and establishes explicitly the connexion between these two types of geometry．In Chapter 3，the idea of isomorphism is carried over from vector spaces to affine．and projective geometries．In particular．we include a simple proof of the basic theorem of projective geometry，in§3．5．This chapter is also the one in which systems of linear equations make their first appearance（§3．3)．They reappear in increasingly sophisticated forms in§§4．5 and 4．6. Linear algebra proper is continued in Chapter 4 with the usua！topics centred on linear mappings．In this chapter the important concept of duality in vector spaces is！inked to the idea of dual geometries．In our treatment of bilinear forms in Chapter 5 we take the theory up to，and including．the classification of symmetric forms over the complex and real fields．
《线性几何（第2版）（英文版）》内容为Vector Spaces、Sets、Groups. Fields and Vector Spaces、Subspaces、Dimension、The Ground Field、Affine and Projective Geometry、Affine Geometries、Affine Propositions of Incidence、Affine lsomorphisms、Homogeneous Vectors、Projective Geometries、The Embedding of Affine Geometry in Projective Geometry、The Fundamental Incidence Theorems of pojective Geometry、Isomorphisms、ffinities、Projectivities等等。
Guide to the ReaderChapter Ⅰ Vector Spaces 1.1 Sets 1.2 Groups, Fields and Vector Spaces 1.3 Subspaces 1.4 Dimension 1.5 The Ground FieldChapter Ⅱ Affine and Projective Geometry 2.1 Affine Geometries 2.2 Affine Propositions of Incidence 2.3 Affine Isomorphisms 2.4 Homogeneous Vectors 2.5 Projective Geometrics 2.6 The Embedding of Affine Geometry in Projective Geometry 2.7 The Fundamental Incidence Theorems of Projective GeometryChapter Ⅲ Isomorphisms 3.1 Affinities 3.2 Projectivities 3.3 Linear Equations 3.4 Affine and Projective Isomorphisms 3.5 Semi-linear Isomorphisms 3.6 Groups of Automorphisms 3.7 Central CollineationsChapter Ⅳ Linear Mappings 4.1 Elementary Properties of Linear Mappings 4.2 Degenerate Affinities and Projectivities 4.3 Matrices 4.4 The Rank of a Linear Mapping 4.5 Linear Equations 4.6 Dual Spaces 4.7 Dualities 4.8 Dual GeometriesChapter Ⅴ Bilinear Forms 5.1 Elementary Properties of Bilinear Forms 5.2 Orthogonality 5.3 Symmetric and Alternating Bilinear Forms 5.4 Structure Theorems 5.5 Correlations 5.6 Projective Quadrics 5.7 Affine Quadrics 5.8 Sesquilinear FormsChapter Ⅵ Euclidean Geometry 6.1 Distances and Euclidean Geometries 6.2 Similarity Euclidean Geometries 6.3 Euclidean Quadrics 6.4 Euclidean Automorphisms 6.5 Hilbert SpacesChapter Ⅶ Modules 7.1 Rings and Modules 7.2 Submodules and Homomorphisms 7.3 Direct Decompositions 7.4 Equivalence of Matrices over F[X] 7.5 Similarity of Matrices over F 7.6 Classification of CoilineationsSolutionsList of SymbolsBibliographyIndex
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线性几何包括仿射几何和射影几何，这部分内容在国内的传统数学教学中长期受到了忽视，然而几何学的重要性已经日趋受人重视。此书融线性代数解析几何为一体，数形结合，更有利于学生弄懂而且弄透。此书内容精练，习题丰富，很适合数学专业的本科生使用，为进入现代数学做好铺垫。