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Linear Algebra (Book): Difference between revisions

Linear Algebra (Book) (edit)

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! colspan="3" | Chapter 6: THE CANONICAL FORM OF THE MATRIX OF A LINEAR OPERATOR
! colspan="3" | Chapter 6: THE CANONICAL FORM OF THE MATRIX OF A LINEAR OPERATOR
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| 1 || Some standard mappings of the plane || 133
| 6.1 || Canonical Form of the Matrix of a Nilpotent Operator || 133
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| 2 || Isometries || 143
| 6.2 || Algebras. The Algebra of Polynomials || 136
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| 3 || Composition of isometries || 150
| 6.3 || Canonical Form of the Matrix of an Arbitrary Operator || 142
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| 4 || Inverse of isometries || 155
| 6.4 || Elementary Divisors || 147
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| 5 || Characterization of isometries || 163
| 6.5 || Further Implications || 153
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| 6 || Congruences || 166
| 6.6 || The Real Jordan Canonical Form || 155
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! colspan="3" | Chapter 7: Area and Applications
| 6.7 || Spectra, Jets and Polynomials || 160
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| 1 || Area of a disc of radius ''r'' || 173
| 6.8 || Operator Functions and Their Matrices || 169
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| 2 || Circumference of a circle of radius ''r'' || 180
| || Problems || 176
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! colspan="3" | PART III: COORDINATE GEOMETRY
! colspan="3" | Chapter 7: BILINEAR AND QUADRATIC FORMS
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! colspan="3" | Chapter 8: Coordinates and Geometry
| 7.1 || Bilinear Forms || 179
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| 1 || Coordinate systems || 191
| 7.2 || Quadratic Forms || 183
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| 2 || Distance between points || 197
| 7.3 || Reduction of a Quadratic Form to Canonical Form || 183
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| 3 || Equation of a circle || 203
| 7.4 || The Canonical Basis of a Bilinear Form || 183
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| 4 || Rational points on a circle || 206
| 7.5 || Construction of a Canonical Basis by Jacobi's Method || 183
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! colspan="3" | Chapter 9: Operations on Points
| 7.6 || Adjoint Linear Operators || 183
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| 1 || Dilations and reflections || 213
| 7.7 || Isomorphism of Spaces Equipped with a Bilinear Form || 183
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| 2 || Addition, subtraction, and the parallelogram law || 218
| *7.8 || Multilinear Forms || 183
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! colspan="3" | Chapter 10: Segments, Rays, and Lines
| 7.9 || Bilinear and Quadratic Forms in a Real Space || 183
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| 1 || Segments || 229
| || Problems || 210
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| 2 || Rays || 231
! colspan="3" | Chapter 8: EUCLIDEAN SPACES
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| 3 || Lines || 236
| 8.1 || Introduction || 214
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| 4 || Ordinary equation for a line || 246
| 8.2 || Definition of a Euclidean Space || 215
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! colspan="3" | Chapter 11: Trigonometry
| 8.3 || Basic Metric Concepts || 216
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| 1 || Radian measure || 249
| 8.4 || Orthogonal Bases || 222
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| 2 || Sine and cosine || 252
| 8.5 || Perpendiculars || 223
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| 3 || The graphs || 264
| 8.6 || The Orthogonalization Theorem || 226
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| 4 || The tangent || 266
| 8.7 || The Gram Determinant || 230
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| 5 || Addition formulas || 272
| 8.8 || Incompatible Systems and the Method of Least Squares || 234
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| 6 || Rotations || 277
| 8.9 || Adjoint Operators and Isometry || 237
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! colspan="3" | Chapter 12: Some Analytic Geometry
| || Problems || 241
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| 1 || The straight line again || 281
! colspan="3" | Chapter 9: UNITARY SPACES
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| 2 || The parabola || 291
| 9.1 || Hermitian Forms || 247
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| 3 || The ellipse || 297
| 9.2 || The Scalar Product in a Complex Space || 254
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| 4 || The hyperbola || 300
| 9.3 || Normal Operators || 259
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| 5 || Rotation of hyperbolas || 305
| 9.4 || Applications to Operator Theory in Euclidean Space || 263
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! colspan="3" | PART IV: MISCELLANEOUS
| || Problems || 271
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! colspan="3" | Chapter 13: Functions
! colspan="3" | Chapter 10: QUADRATIC FORMS IN EUCLIDEAN AND UNITARY SPACES
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| 1 || Definition of a function || 313
| 10.1 || Basic Theorem on Quadratic Forms in a Euclidean Space || 273
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| 2 || Polynomial functions || 318
| 10.2 || Extremal Properties of a Quadratic Form || 276
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| 3 || Graphs of functions || 330
| 10.3 || Simultaneous Reduction of Two Quadratic Forms || 283
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| 4 || Exponential function || 333
| 10.4 || Reduction of the General Equation of a Quadric Surface || 287
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| 5 || Logarithms || 338
| 10.5 || Geometric Properties of a Quadric Surface || 289
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! colspan="3" | Chapter 14: Mappings
| *10.6 || Analysis of a Quadric Surface from Its General Equation || 300
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| 1 || Definition || 345
| 10.7 || Hermitian Quadratic Forms || 308
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| 2 || Formalism of mappings || 351
| || Problems || 310
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| 3 || Permutations || 359
! colspan="3" | Chapter 11: FINITE-DIMENSIONAL ALGEBRAS AND THEIR REPRESENTATIONS
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! colspan="3" | Chapter 15: Complex Numbers
| 11.1 || More on Algebras || 312
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| 1 || The complex plane || 375
| 11.2 || Representations of Abstract Algebras || 313
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| 2 || Polar form || 380
| 11.3 || Irreducible Representations and Schur's Lemma || 314
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! colspan="3" | Chapter 16: Induction and Summations
| 11.4 || Basic Types of Finite-Dimensional Algebras || 315
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| 1 || Induction || 383
| 11.5 || The Left Regular Representation of a Simple Algebra || 318
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| 2 || Summations || 388
| 11.6 || Structure of Simple Algebras || 320
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| 3 || Geometric series || 396
| 11.7 || Structure of Semisimple Algebras || 323
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! colspan="3" | Chapter 17: Determinants
| 11.8 || Representations of Simple and Semisimple Algebras || 327
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| 1 || Matrices || 401
| 11.9 || Some Further Results || 331
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| 2 || Determinants of order 2 || 406
| || Problems || 332
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| 3 || Properties of 2 x 2 determinants || 409
| *Appendix || ||  
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| 4 || Determinants of order 3 || 414
! colspan="3" | CATEGORIES OF FINITE-DIMENSIONAL SPACES
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| 5 || Properties of 3 x 3 determinants || 418
| A.1 || Introduction || 335
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| 6 || Cramer's Rule || 424
| A.2 || The Case of Complete Algebras || 338
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! colspan="2" | Index || 429
| A.3 || The Case of One-Dimensional Algebras || 340
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| A.4 || The Case of Simple Algebras || 345
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| A.5 || The Case of Complete Algebras of Diagonal Matrices || 353
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| A.6 || Categories and Direct Sums || 357
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! colspan="2" | HINTS AND ANSWERS || 361
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! colspan="2" | BIBLIOGRAPHY || 379
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! colspan="2" | INDEX || 381
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[[Category:Mathematics]]
[[Category:Mathematics]]
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