chapter 1 |
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DETERMINANTS |
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1.1. |
Number Fields |
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1.2. |
Problems of the Theory of Systems of Linear Equations |
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1.3. |
Determinants of Order n |
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1.4. |
Properties of Determinants |
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1.5. |
Cofactors and Minors |
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1.6. |
Practical Evaluation of Determinants |
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1.7. |
Cramer's Rule |
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1.8. |
Minors of Arbitrary Order. Laplace's Theorem |
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1.9. |
Linear Dependence between Columns |
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Problems |
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chapter 2 |
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LINEAR SPACES |
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2.1. |
Definitions |
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2.2. |
Linear Dependence |
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2.3. |
"Bases, Components, Dimension" |
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2.4. |
Subspaces |
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2.5. |
Linear Manifolds |
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2.6. |
Hyperplanes |
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2.7. |
Morphisms of Linear Spaces |
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Problems |
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chapter 3 |
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SYSTEMS OF LINEAR EQUATIONS |
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3.1. |
More on the Rank of a Matrix |
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3.2. |
Nontrivial Compatibility of a Homogeneous Linear System |
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3.3. |
The Compatability Condition for a General Linear System |
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3.4. |
The General Solution of a Linear System |
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3.5. |
Geometric Properties of the Solution Space |
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3.6. |
Methods for Calculating the Rank of a Matrix |
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Problems |
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chapter 4 |
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LINEAR FUNCTIONS OF A VECTOR ARGUMENT |
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4.1. |
Linear Forms |
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4.2. |
Linear Operators |
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4.3. |
Sums and Products of Linear Operators |
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4.4. |
Corresponding Operations on Matrices |
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4.5. |
Further Properties of Matrix Multiplication |
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4.6. |
The Range and Null Space of a Linear Operator |
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4.7. |
Linear Operators Mapping a Space Kn into Itself |
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4.8. |
Invariant Subspaces |
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4.9. |
Eigenvectors and Eigenvalues |
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Problems |
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chapter 5 |
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COORDINATE TRANSFORMATIONS |
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5.1. |
Transformation to a New Basis |
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5.2. |
Consecutive Transformations |
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5.3. |
Transformation of the Components of a Vector |
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5.4. |
Transformation of the Coefficients of a Linear Form |
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5.5. |
Transformation of the Matrix of a Linear Operator |
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*5.6. |
Tensors |
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Problems |
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chapter 6 |
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THE CANONICAL FORM OF THE MATRIX OF A LINEAR OPERATOR |
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6.1. |
Canonical Form of the Matrix of a Nilpotent Operator |
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6.2. |
Algebras. The Algebra of Polynomials |
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6.3. |
Canonical Form of the Matrix of an Arbitrary Operator |
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6.4. |
Elementary Divisors |
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6.5. |
Further Implications |
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6.6. |
The Real Jordan Canonical Form |
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*6.7. |
"Spectra, Jets and Polynomials" |
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*6.8. |
Operator Functions and Their Matrices |
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Problems |
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chapter 7 |
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BILINEAR AND QUADRATIC FORMS |
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7.1. |
Bilinear Forms |
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7.2. |
Quadratic Forms |
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7.3. |
Reduction of a Quadratic Form to Canonical Form |
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7.4. |
The Canonical Basis of a Bilinear Form |
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7.5. |
Construction of a Canonical Basis by Jacobi's Method |
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7.6. |
Adjoint Linear Operators |
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7.7. |
Isomorphism of Spaces Equipped with a Bilinear Form |
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*7.8. |
Multilinear Forms |
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7.9. |
Bilinear and Quadratic Forms in a Real Space |
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Problems |
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chapter 8 |
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EUCLIDEAN SPACES |
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8.1. |
Introduction |
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8.2. |
Definition of a Euclidean Space |
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8.3. |
Basic Metric Concepts |
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8.4. |
Orthogonal Bases |
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8.5. |
Perpendiculars |
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8.6. |
The Orthogonalization Theorem |
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8.7. |
The Gram Determinant |
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8.8. |
Incompatible Systems and the Method of Least Squares |
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8.9. |
Adjoint Operators and Isometry |
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Problems |
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chapter 9 |
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UNITARY SPACES |
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9.1. |
Hermitian Forms |
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9.2. |
The Scalar Product in a Complex Space |
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9.3. |
Normal Operators |
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9.4. |
Applications to Operator Theory in Euclidean Space |
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Problems |
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chapter 10 |
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QUADRATIC FORMS IN EUCLIDEAN AND UNITARY SPACES |
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10.1. |
Basic Theorem on Quadratic Forms in a Euclidean Space |
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10.2. |
Extremal Properties of a Quadratic Form |
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10.3 |
Simultaneous Reduction of Two Quadratic Forms |
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10.4. |
Reduction of the General Equation of a Quadratic Surface |
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10.5. |
Geometric Properties of a Quadratic Surface |
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*10.6. |
Analysis of a Quadric Surface from Its Genearl Equation |
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10.7. |
Hermitian Quadratic Forms |
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Problems |
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chapter 11 |
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FINITE-DIMENSIONAL ALGEBRAS AND THEIR REPRESENTATIONS |
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11.1. |
More on Algebras |
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11.2. |
Representations of Abstract Algebras |
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11.3. |
Irreducible Representations and Schur's Lemma |
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11.4. |
Basic Types of Finite-Dimensional Algebras |
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11.5. |
The Left Regular Representation of a Simple Algebra |
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11.6. |
Structure of Simple Algebras |
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11.7. |
Structure of Semisimple Algebras |
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11.8. |
Representations of Simple and Semisimple Algebras |
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11.9. |
Some Further Results |
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Problems |
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*Appendix |
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CATEGORIES OF FINITE-DIMENSIONAL SPACES |
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A.1. |
Introduction |
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A.2. |
The Case of Complete Algebras |
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A.3. |
The Case of One-Dimensional Algebras |
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A.4. |
The Case of Simple Algebras |
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A.5. |
The Case of Complete Algebras of Diagonal Matrices |
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A.6. |
Categories and Direct Sums |
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HINTS AND ANSWERS |
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BIBLIOGRAPHY |
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INDEX |