Linear Algebra

1 Sets and Functions. - 1. 1 Notation and Terminology. - 1. 2 Composition of Functions. - 1. 3 Inverse Functions. - 1. 4 Digression on Cardinality. - 1. 5 Permutations. - Exercises. - 2 Groups and Group Homomorphisms. - 2. 1 Groups and Subgroups. - 2. 2 Group Homomorphisms. - 2. 3 Rings and Fields. - Exercises. - 3 Vector Spaces and Linear Transformations. - 3. 1 Vector Spaces and Subspaces. - 3. 2 Linear Transformations. - 3. 3 Direct Products and Internal Direct Sums. - Exercises. - 4 Dimension. - 4. 1 Bases and Dimension. - 4. 2 Vector Spaces Are Free. - 4. 3 Rank and Nullity. - Exercises. - 5 Matrices. - 5. 1 Notation and Terminology. - 5. 2 Introduction to Linear Systems. - 5. 3 Solution Techniques. - 5. 4 Multiple Systems and Matrix Inversion. - Exercises. - 6 Representation of Linear Transformations. - 6. 1 The Space of Linear Transformations. - 6. 2 The Representation of Hom(knkm). - 6. 3 The Representation of Hom(VV'). - 6. 4 The Dual Space. - 6. 5 Change of Basis. - Exercises. - 7 Inner Product Spaces. - 7. 1 Real Inner Product Spaces. - 7. 2 Orthogonal Bases and Orthogonal Projection. - 7. 3 Complex Inner Product Spaces. - Exercises. - 8 Determinants. - 8. 1 Existence and Basic Properties. - 8. 2 A Nonrecursive Formula; Uniqueness. - 8. 3 The Determinant of a Product; Invertibility. - Exercises. - 9 Eigenvalues and Eigenvectors. - 9. 1 Definitions and Elementary Properties. - 9. 2 Hermitian and Unitary Transformations. - 9. 3 Spectral Decomposition. - Exercises. - 10 Triangulation and Decomposition of Endomorphisms. - 10. 1 The Cayley-Hamilton Theorem. - 10. 2 Triangulation of Endomorphisms. - 10. 3 Decomposition by Characteristic Subspaces. - 10. 4 Nilpotent Mappings and the Jordan Normal Form. - Exercises. - Supplementary Topics. - 1 Differentiation. - 2 The Determinant Revisited. - 3 Quadratic Forms. - 4 An Introduction to Categories and Functors. Language: English