Linear Algebra

1. Vectors in the Plane and in Space. - 1. 1 First Steps. - 1. 2 Exercises. - 2. Vector Spaces. - 2. 1 Axioms for Vector Spaces. - 2. 2 Cartesian (or Euclidean) Spaces. - 2. 3 Some Rules for Vector Algebra. - 2. 4 Exercises. - 3. Examples of Vector Spaces. - 3. 1 Three Basic Examples. - 3. 2 Further Examples of Vector Spaces. - 3. 3 Exercises. - 4. Subspaces. - 4. 1 Basic Properties of Vector Subspaces. - 4. 2 Examples of Subspaces. - 4. 3 Exercises. - 5. Linear Independence and Dependence. - 5. 1 Basic Definitions and Examples. - 5. 2 Properties of Independent and Dependent Sets. - 5. 3 Exercises. - 6. Finite-Dimensional Vector Spaces and Bases. - 6. 1 Finite-Dimensional Vector Spaces. - 6. 2 Properties of Bases. - 6. 3 Using Bases. - 6. 4 Exercises. - 7. The Elements of Vector Spaces: A Summing Up. - 7. 1 Numerical Examples. - 7. 2 Exercises. - 8. Linear Transformations. - 8. 1 Definition of Linear Transformations. - 8. 2 Examples of Linear Transformations. - 8. 3 Properties of Linear Transformations. - 8. 4 Images and Kernels of Linear Transformations. - 8. 5 Some Fundamental Constructions. - 8. 6 Isomorphism of Vector Spaces. - 8. 7 Exercises. - 9. Linear Transformations: Examples and Applications. - 9. 1 Numerical Examples. - 9. 2 Some Applications. - 9. 3 Exercises. - 10. Linear Transformations and Matrices. - 10. 1 Linear Transformations and Matrices in IR3. - 10. 2 Some Numerical Examples. - 10. 3 Matrices and Their Algebra. - 10. 4 Special Types of Matrices. - 10. 5 Exercises. - 11. Representing Linear Transformations by Matrices. - 11. 1 Representing a Linear Transformation by a Matrix. - 11. 2 Basic Theorems. - 11. 3 Change of Bases. - 11. 4 Exercises. - 12. More on Representing Linear Transformations by Matrices. - 12. 1 Projections. - 12. 2 Nilpotent Transformations. - 12. 3 Cyclic Transformations. - 12. 4 Exercises. - 13. Systems of Linear Equations. - 13. 1 Existence Theorems. - 13. 2 Reduction to Echelon Form. - 13. 3 The Simplex Method. - 13. 4 Exercises. - 14. The Elements of Eigenvalue and Eigenvector Theory. - 14. 1 The Rank of an Endomorphism. - 14. 2 Eigenvalues and Eigenvectors. - 14. 3 Determinants. - 14. 4 The Characteristic Polynomial. - 14. 5 Diagonalization Theorems. - 14. 6 Exercises. - 15. Inner Product Spaces. - 15. 1 Scalar Products. - 15. 2 Inner Product Spaces. - 15. 3 Isometries. - 15. 4 The Riesz Representation Theorem. - 15. 5 Legendre Polynomials. - 15. 6 Exercises. - 16. The Spectral Theorem and Quadratic Forms. - 16. 1 Self-Adjoint Transformations. - 16. 2 The Spectral Theorem. - 16. 3 The Principal Axis Theorem for Quadratic Forms. - 16. 4 A Proof of the Spectral Theorem in the General Case. - 16. 5 Exercises. - 17. Jordan Canonical Form. - 17. 1 Invariant Subspaces. - 17. 2 Nilpotent Transformations. - 17. 3 The Jordan Normal Form. - 17. 4 Square Roots. - 17. 5 The Hamilton-Cayley Theorem. - 17. 6 Inverses. - 17. 7 Exercises. - 18. Application to Differential Equations. - 18. 1 Linear Differential Systems: Basic Definitions. - 18. 2 Diagonalizable Systems. - 18. 3 Application of Jordan Form. - 18. 4 Exercises. - 19. The Similarity Problem. - 19. 1 The Fundamental Problem ofLinear Algebra. - 19. 2 A Bit of Invariant Theory. - 19. 3 Exercises. - A. Multilinear Algebra and Determinants. - A. 1 Multilinear Forms. - A. 2 Determinants. - A. 3 Exercises. - B. Complex Numbers. - B. 1 The Complex Numbers. - B. 2 Exercises. - Font Usage. - Notations. Language: English