Editor's Preface | p. iii |
Translators' Preface | p. iii |
Authors' Preface | p. v |
Affine Space; Linear Equations | |
n-dimensional Affine Space | p. 1 |
Vectors | p. 6 |
The Concept of Linear Dependence | p. 16 |
Vector Spaces in Rn | p. 19 |
Linear Spaces | p. 27 |
Linear Equations | p. 34 |
Homogeneous Linear Equations | p. 36 |
Non-homogeneous Linear Equations | p. 40 |
Geometric Applications | p. 44 |
Euclidean Space; Theory of Determinants | |
Euclidean Length | p. 50 |
Calculating with the Summation Sign | p. 60 |
Volumes and Determinants | p. 63 |
Fundamental Properties of Determinants | p. 69 |
Existence and Uniqueness of Determinants | p. 74 |
Volumes | p. 83 |
The Principal Theorems of Determinant Theory | p. 87 |
The Complete Development of a Determinant | p. 87 |
The Determinant as a Function of its Column Vectors | p. 89 |
The Multiplication Theorem | p. 96 |
The Development of a Determinant by Rows or Columns | p. 98 |
Determinants and Linear Equations | p. 100 |
Laplace's Expansion Theorem | p. 105 |
Transformation of Coordinates | p. 117 |
General Linear Coordinate Systems | p. 117 |
Cartesian Coordinate Systems | p. 126 |
Continuous Deformation of a Linear Coordinate System | p. 131 |
Construction of Normal Orthogonal Systems and Applications | p. 140 |
Rigid Motions | p. 153 |
Rigid Motions in R2 | p. 162 |
Rigid Motions in R3 | p. 168 |
Affine Transformations | p. 180 |
Field Theory; The Fundamental Theorem of Algebra | |
The Concept of a Field | p. 187 |
Polynomials over a Field | p. 204 |
The Field of Complex Numbers | p. 218 |
The Fundamental Theorem of Algebra | p. 230 |
Elements of Group Theory | |
The Concept of a Group | p. 245 |
Subgroups; Examples | p. 251 |
The Basis Theorem for Abelian Groups | p. 260 |
Linear Transformations and Matrices | |
The Algebra of Linear Transformations | p. 273 |
Calculation with Matrices | p. 283 |
Linear Transformations Under a Change of Coordinate System | p. 293 |
The Determinant of a Linear Transformations | p. 296 |
Linear Dependence of Matrices | p. 297 |
Calculation With Matrix Polynomials | p. 298 |
The Transpose of a Matrix | p. 301 |
The Minimal Polynomial; Invariant Subspaces | p. 303 |
The Minimal Polynomial | p. 303 |
Invariant Subspaces | p. 305 |
The Nullspace of a Linear Transformation f(¿) | p. 306 |
Decomposition of L into Invariant Subspaces | p. 310 |
Geometric Interpretation | p. 315 |
The Diagonal Form and its Applications | p. 320 |
Unitary Transformations | p. 327 |
Orthogonal Transformations | p. 334 |
Hermitian and Symmetric Matrices (Principal Axis Transformations) | p. 340 |
The Elementary Divisors of a Polynomial Matrix | p. 344 |
The Normal Form | p. 355 |
Consequences | p. 363 |
Linear Transformation with Prescribed Elementary Divisors | p. 365 |
The Jordan Normal Form | p. 367 |
Index | p. 373 |
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