Algebra and GeometryCambridge University Press, 2005-05-12 Describing two cornerstones of mathematics, this basic textbook presents a unified approach to algebra and geometry. It covers the ideas of complex numbers, scalar and vector products, determinants, linear algebra, group theory, permutation groups, symmetry groups and aspects of geometry including groups of isometries, rotations, and spherical geometry. The book emphasises the interactions between topics, and each topic is constantly illustrated by using it to describe and discuss the others. Many ideas are developed gradually, with each aspect presented at a time when its importance becomes clearer. To aid in this, the text is divided into short chapters, each with exercises at the end. The related website features an HTML version of the book, extra text at higher and lower levels, and more exercises and examples. It also links to an electronic maths thesaurus, giving definitions, examples and links both to the book and to external sources. |
Turinys
| 1 | |
2 The real numbers | 22 |
3 The complex plane | 31 |
4 Vectors in threedimensional space | 52 |
5 Spherical geometry | 74 |
6 Quaternions and isometries | 89 |
7 Vector spaces | 102 |
8 Linear equations | 135 |
10 Eigenvectors | 175 |
11 Linear maps of Euclidean space | 197 |
12 Groups | 215 |
13 Möbius transformations | 254 |
14 Group actions | 284 |
15 Hyperbolic geometry | 307 |
| 320 | |
9 Matrices | 149 |
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abelian algebraic angle basis bijection circle complex numbers composition consider cosets cross-ratio cyclic group deduce defined Definition denote det(A diagonal dihedral group dim(U dim(V dimension direct isometry edges eigenvalues eigenvectors element of G equation equivalent Euclidean example Exercise f and g finite fixed points frieze group function g in G geometry given group G group of order group with respect hence homomorphism hyperbolic integer inverse isomorphic kernel Lemma Let G linear combination linear map linearly independent M¨obius map matrix representation Möbius map Möbius transformation non-zero normal subgroup orthogonal orthogonal matrix permutation plane polyhedron polynomial Proof Let proof of Theorem prove quaternions real numbers reflection rotation scalar multiple scalar product Show solution span spherical subgroup of G subspace Suppose surjective symmetry group transpositions triangle unique vector product vector space vertex vertices zero