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(kn,km).- 6.3 The Representation of Hom(V,V').- 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.