Linear Algebra

chapter 1 DETERMINANTS 1.1. Number Fields 1.2. Problems of the Theory of Systems of Linear Equations 1.3. Determinants of Order n 1.4. Properties of Determinants 1.5. Cofactors and Minors 1.6. Practical Evaluation of Determinants 1.7. Cramer's Rule 1.8. Minors of Arbitrary Order. Laplace's Theorem 1.9. Linear Dependence between ColumnsProblemschapter 2 LINEAR SPACES 2.1. Definitions 2.2. Linear Dependence 2.3. "Bases, Components, Dimension" 2.4. Subspaces 2.5. Linear Manifolds 2.6. Hyperplanes 2.7. Morphisms of Linear SpacesProblemschapter 3 SYSTEMS OF LINEAR EQUATIONS 3.1. More on the Rank of a Matrix 3.2. Nontrivial Compatibility of a Homogeneous Linear System 3.3. The Compatability Condition for a General Linear System 3.4. The General Solution of a Linear System 3.5. Geometric Properties of the Solution Space 3.6. Methods for Calculating the Rank of a MatrixProblemschapter 4 LINEAR FUNCTIONS OF A VECTOR ARGUMENT 4.1. Linear Forms 4.2. Linear Operators 4.3. Sums and Products of Linear Operators 4.4. Corresponding Operations on Matrices 4.5. Further Properties of Matrix Multiplication 4.6. The Range and Null Space of a Linear Operator 4.7. Linear Operators Mapping a Space Kn into Itself 4.8. Invariant Subspaces 4.9. Eigenvectors and EigenvaluesProblemschapter 5 COORDINATE TRANSFORMATIONS 5.1. Transformation to a New Basis 5.2. Consecutive Transformations 5.3. Transformation of the Components of a Vector 5.4. Transformation of the Coefficients of a Linear Form 5.5. Transformation of the Matrix of a Linear Operator *5.6. TensorsProblemschapter 6 THE CANONICAL FORM OF THE MATRIX OF A LINEAR OPERATOR 6.1. Canonical Form of the Matrix of a Nilpotent Operator 6.2. Algebras. The Algebra of Polynomials 6.3. Canonical Form of the Matrix of an Arbitrary Operator 6.4. Elementary Divisors 6.5. Further Implications 6.6. The Real Jordan Canonical Form *6.7. "Spectra, Jets and Polynomials" *6.8. Operator Functions and Their MatricesProblemschapter 7 BILINEAR AND QUADRATIC FORMS 7.1. Bilinear Forms 7.2. Quadratic Forms 7.3. Reduction of a Quadratic Form to Canonical Form 7.4. The Canonical Basis of a Bilinear Form 7.5. Construction of a Canonical Basis by Jacobi's Method 7.6. Adjoint Linear Operators 7.7. Isomorphism of Spaces Equipped with a Bilinear Form *7.8. Multilinear Forms 7.9. Bilinear and Quadratic Forms in a Real SpaceProblemschapter 8 EUCLIDEAN SPACES 8.1. Introduction 8.2. Definition of a Euclidean Space 8.3. Basic Metric Concepts 8.4. Orthogonal Bases 8.5. Perpendiculars 8.6. The Orthogonalization Theorem 8.7. The Gram Determinant 8.8. Incompatible Systems and the Method of Least Squares 8.9. Adjoint Operators and IsometryProblemschapter 9 UNITARY SPACES 9.1. Hermitian Forms 9.2. The Scalar Product in a Complex Space 9.3. Normal Operators 9.4. Applications to Operator Theory in Euclidean SpaceProblemschapter 10 QUADRATIC FORMS IN EUCLIDEAN AND UNITARY SPACES 10.1. Basic Theorem on Quadratic Forms in a Euclidean Space 10.2. Extremal Properties of a Quadratic Form 10.3 Simultaneous Reduction of Two Quadratic Forms 10.4. Reduction of the General Equation of a Quadratic Surface 10.5. Geometric Properties of a Quadratic Surface *10.6. Analysis of a Quadric Surface from Its Genearl Equation10.7. Hermitian Quadratic FormsProblemschapter 11 FINITE-DIMENSIONAL ALGEBRAS AND THEIR REPRESENTATIONS 11.1. More on Algebras 11.2. Representations of Abstract Algebras 11.3. Irreducible Representations and Schur's Lemma 11.4. Basic Types of Finite-Dimensional Algebras 11.5. The Left Regular Representation of a Simple Algebra 11.6. Structure of Simple Algebras 11.7. Structure of Semisimple Algebras 11.8. Representations of Simple and Semisimple Algebras 11.9. Some Further ResultsProblems*Appendix CATEGORIES OF FINITE-DIMENSIONAL SPACES A. 1. Introduction A. 2. The Case of Complete Algebras A. 3. The Case of One-Dimensional Algebras A. 4. The Case of Simple Algebras A. 5. The Case of Complete Algebras of Diagonal Matrices A. 6. Categories and Direct SumsHINTS AND ANSWERSBIBLIOGRAPHYINDEX