Editing Linear Algebra (Book) (section)

== Table of Contents ==

{| class="wikitable"
|-
! Chapter/Section # !! Title !! Page #
|- 
! colspan="3" | Chapter 1: DETERMINANTS
|-
| 1.1 || Number Fields || 1
|-
| 1.2 || Problems of the Theory of Systems of Linear Equations || 3
|-
| 1.3 || Determinants of Order <math>n</math> || 5
|-
| 1.4 || Properties of Determinants || 8
|-
| 1.5 || Cofactors and Minors || 12
|-
| 1.6 || Practical Evaluation of Determinants || 16
|- 
| 1.7 || Cramer's Rule || 18
|- 
| 1.8 || Minors of Arbitrary Order. Laplace's Theorem || 20
|- 
| 1.9 || Multiplicative inverses || 23
|- 
| || Problems || 28
|- 
! colspan="3" | Chapter 2: LINEAR SPACES
|-
| 2.1 || Definitions || 31
|-
| 2.2 || Linear Dependence || 36
|-
| 2.3 || Bases, Components, Dimension || 38
|-
| 2.4 || Subspaces || 42
|-
| 2.5 || Linear Manifolds || 49
|-
| 2.6 || Hyperplanes || 51
|-
| 2.7 || Morphisms of Linear Spaces || 53
|-
| || Problems || 56
|-
! colspan="3" | Chapter 3: SYSTEMS OF LINEAR EQUATIONS
|-
| 3.1 || More on the Rank of a Matrix || 58
|-
| 3.2 || Nontrivial Compatibility of a Homogeneous Linear System || 60
|-
| 3.3 || The Compatibility Condition for a General Linear System || 61
|-
| 3.4 || The General Solution of a Linear System || 63
|-
| 3.4 || Geometric Properties of the Solution Space || 65
|-
| 3.4 || Methods for Calculating the Rank of a Matrix || 67
|-
|  || Problems || 71
|-
! colspan="3" | Chapter 4: LINEAR FUNCTIONS OF A VECTOR ARGUMENT
|-
| 4.1 || Linear Forms || 75
|-
| 4.2 || Linear Operators || 77
|-
| 4.3 || Sums and Products of Linear Operators || 82
|-
| 4.4 || Corresponding Operations on Matrices || 84
|-
| 4.5 || Further Properties of Matrix Multiplication || 88
|-
| 4.6 || The Range and Null Space of a Linear Operator || 93
|-
| 4.7 || Linear Operators Mapping a Space <math>K_n</math> into Itself || 98
|-
| 4.8 || Invariant Subspaces || 106
|-
| 4.9 || Eigenvectors and Eigenvalues || 108
|-
| || Problems|| 113
|-
! colspan="3" | Chapter 5: COORDINATE TRANSFORMATIONS
|-
| 5.1 || Transformation to a New Basis || 118
|-
| 5.2 || Consecutive Transformations || 120
|-
| 5.3 || Transformation of the Components of a Vector  || 121
|-
| 5.4 || Transformation of the Coefficients of a Linear Form || 123
|-
| 5.5 || Transformation of the Matrix of a Linear Operator || 124
|-
| *5.6 || Tensors || 126
|-
| || Problems || 131
|-
! colspan="3" | Chapter 6: THE CANONICAL FORM OF THE MATRIX OF A LINEAR OPERATOR
|-
| 6.1 || Canonical Form of the Matrix of a Nilpotent Operator || 133
|-
| 6.2 || Algebras. The Algebra of Polynomials || 136
|-
| 6.3 || Canonical Form of the Matrix of an Arbitrary Operator || 142
|-
| 6.4 || Elementary Divisors || 147
|-
| 6.5 || Further Implications || 153
|-
| 6.6 || The Real Jordan Canonical Form || 155
|-
| 6.7 || Spectra, Jets and Polynomials || 160
|-
| 6.8 || Operator Functions and Their Matrices || 169
|-
| || Problems || 176
|-
! colspan="3" | Chapter 7: BILINEAR AND QUADRATIC FORMS
|-
| 7.1 || Bilinear Forms || 179
|-
| 7.2 || Quadratic Forms || 183
|-
| 7.3 || Reduction of a Quadratic Form to Canonical Form || 183
|-
| 7.4 || The Canonical Basis of a Bilinear Form || 183
|-
| 7.5 || Construction of a Canonical Basis by Jacobi's Method || 183
|-
| 7.6 || Adjoint Linear Operators || 183
|-
| 7.7 || Isomorphism of Spaces Equipped with a Bilinear Form || 183
|-
| *7.8 || Multilinear Forms || 183
|-
| 7.9 || Bilinear and Quadratic Forms in a Real Space || 183
|-
| || Problems || 210
|-
! colspan="3" | Chapter 8: EUCLIDEAN SPACES
|-
| 8.1 || Introduction || 214
|-
| 8.2 || Definition of a Euclidean Space || 215
|-
| 8.3 || Basic Metric Concepts || 216
|-
| 8.4 || Orthogonal Bases || 222
|-
| 8.5 || Perpendiculars || 223
|-
| 8.6 || The Orthogonalization Theorem || 226
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| 8.7 || The Gram Determinant || 230
|-
| 8.8 || Incompatible Systems and the Method of Least Squares || 234
|-
| 8.9 || Adjoint Operators and Isometry || 237
|-
| || Problems || 241
|-
! colspan="3" | Chapter 9: UNITARY SPACES
|-
| 9.1 || Hermitian Forms || 247
|-
| 9.2 || The Scalar Product in a Complex Space || 254
|-
| 9.3 || Normal Operators || 259
|-
| 9.4 || Applications to Operator Theory in Euclidean Space || 263
|-
| || Problems || 271
|-
! colspan="3" | Chapter 10: QUADRATIC FORMS IN EUCLIDEAN AND UNITARY SPACES
|-
| 10.1 || Basic Theorem on Quadratic Forms in a Euclidean Space || 273
|-
| 10.2 || Extremal Properties of a Quadratic Form || 276
|-
| 10.3 || Simultaneous Reduction of Two Quadratic Forms || 283
|-
| 10.4 || Reduction of the General Equation of a Quadric Surface || 287
|-
| 10.5 || Geometric Properties of a Quadric Surface || 289
|-
| *10.6 || Analysis of a Quadric Surface from Its General Equation || 300
|-
| 10.7 || Hermitian Quadratic Forms || 308
|-
| || Problems || 310
|-
! colspan="3" | Chapter 11: FINITE-DIMENSIONAL ALGEBRAS AND THEIR REPRESENTATIONS
|-
| 11.1 || More on Algebras || 312
|-
| 11.2 || Representations of Abstract Algebras || 313
|-
| 11.3 || Irreducible Representations and Schur's Lemma || 314
|-
| 11.4 || Basic Types of Finite-Dimensional Algebras || 315
|-
| 11.5 || The Left Regular Representation of a Simple Algebra || 318
|-
| 11.6 || Structure of Simple Algebras || 320
|-
| 11.7 || Structure of Semisimple Algebras || 323
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| 11.8 || Representations of Simple and Semisimple Algebras || 327
|-
| 11.9 || Some Further Results || 331
|-
| || Problems || 332
|-
| *Appendix || || 
|-
! colspan="3" | CATEGORIES OF FINITE-DIMENSIONAL SPACES
|-
| A.1 || Introduction || 335
|-
| A.2 || The Case of Complete Algebras || 338
|-
| A.3 || The Case of One-Dimensional Algebras || 340
|-
| A.4 || The Case of Simple Algebras || 345
|-
| A.5 || The Case of Complete Algebras of Diagonal Matrices || 353
|-
| A.6 || Categories and Direct Sums || 357
|-
! colspan="2" | HINTS AND ANSWERS || 361
|-
! colspan="2" | BIBLIOGRAPHY || 379
|-
! colspan="2" | INDEX || 381
|-
|}

[[Category:Mathematics]]
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