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Bibliografická citace

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BK
Berlin ; Heidelberg : Springer, [2013]
xxi, 526 stran : ilustrace ; 25 cm

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ISBN 978-3-642-30993-9 (vázáno)
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Obsahuje bibliografické odkazy a rejstřík
001463023
Contents // 1 Linear Equations... // 1.1 Linear Equations and Functions... 1 // 1.2 Gaussian Elimination... 6 // 1.3 Examples*... 15 // 2 Matrices and Determinants... 25 // 2.1 Determinants of Orders 2 and 3... 25 // 2.2 Determinants of Arbitrar)’ Order ... 30 // 2.3 Properties that Characterize Determinants... 37 // 2.4 Expansion of a Determinant Along Its Columns... 39 // 2.5 Cramer’s Rule... 42 // 2.6 Permutations, Symmetric and Antisymmetric Functions... 44 // 2.7 Explicit Formula for the Determinant... 50 // 2.8 The Rank of a Matrix... 53 // 2.9 Operations on Matrices... 60 // 2.10 Inverse Matrices... 70 // 3 Vector Spaces... 79 // 3.1 The Definition of a Vector Space... 79 // 3.2 Dimension and Basis ... 86 // 3.3 Linear Transformations of Vector Spaces...101 // 3.4 Change of Coordinates...107 // 3.5 Isomorphisms of Vector Spaces...112 // 3.6 The Rank of a Linear Transformation...118 // 3.7 Dual Spaces...120 // 3.8 Forms and Polynomials in Vectors ...127 // 4 Linear Transformations of a Vector Space to Itself...133 // 4.1 Eigenvectors and Invariant Subspaces...133 // 4.2 Complex and Rea! Vector Spaces...142 // 4.3 Complexification...149 // 4.4 Orientation of a Real Vector Space...154 // vii // viii Contents // 5 Jordan Normal Form...161 // 5.1 Principal Vectors and Cyclic Subspaces...161 // 5.2 Jordan Normal Form (Decomposition)...165 // 5.3 Jordan Normal Form (Uniqueness)...169 // 5.4 Real Vector Spaces...173 // 5.5 Applications* ...176 // 6 Quadratic and Bilinear Forms...191
// 6.1 Basic Definitions...191 // 6.2 Reduction to Canonical Form...198 // 6.3 Complex, Real, and Hermitian Forms...204 // 7 Euclidean Spaces...213 // 7.1 The Definition of a Euclidean Space ...213 // 7.2 Orthogonal Transformations...223 // 7.3 Orientation of a Euclidean Space*...230 // 7.4 Examples*...233 // 7.5 Symmetric Transformations...245 // 7.6 Applications to Mechanics and Geometry*...255 // 7.7 Pseudo-Euclidean Spaces...265 // 7.8 Lorentz Transformations...275 // 8 Affine Spaces...289 // 8.1 The Definition of an Affine Space...289 // 8.2 Alfine Spaces ...294 // 8.3 Affine Transformations...301 // 8.4 Affine Euclidean Spaces and Motions...309 // 9 Projective Spaces...319 // 9.1 Definition of a Projective Space...319 // 9.2 Projective Transformations ...328 // 9.3 The Cross Ratio...335 // 9.4 Topological Properties of Projective Spaces*...339 // 10 The Exterior Product and Exterior Algebras ...349 // 10.1 Pliicker Coordinates of a Subspacc...349 // 10.2 The Pliicker Relations and the Grassmannian...353 // 10.3 The Exterior Product...358 // 10.4 Exterior Algebras*...367 // 10.5 Appendix*...374 // 11 Quadrics...385 // 11.1 Quadrics in Projective Space ...385 // 11.2 Quadrics in Complex Projective Space...394 // 11.3 Isotropic Subspaces...398 // 11.4 Quadrics in a Real Projective Space...410 // 11.5 Quadrics in a Real Affine Space...414 // 11.6 Quadrics in an Affine Euclidean Space...425 // 11.7 Quadrics in the Real Plane*...428 // Contents // 12 Hyperbolic Geometry ...433
// 12.1 Hyperbolic Space*...434 // 12.2 The Axioms of Plane Geometry*...443 // 12.3 Some Formulas of Hyperbolic Geometry*...454 // 13 Groups, Rings, and Modules...467 // 13.1 Groups and Homomorphisms...467 // 13.2 Decomposition of Finite Abelian Groups...475 // 13.3 The Uniqueness of the Decomposition...481 // 13.4 Finitely Generated Torsion Modules over a Euclidean Ring* ... 484 // 14 Elements of Representation Theory...497 // 14.1 Basic Concepts of Representation Theory...497 // 14.2 Representations of Finite Groups...503 // 14.3 Irreducible Representations...508 // 14.4 Representations of Abelian Groups...511 // Historical Note...515 // References...517 // Index...521

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