Linear Algebra Illustrated

Preface Sets and functions Introduction to linear algebra Sets The real number line Set building Relations The xy-plane Functions Operations on sets Linear relations and functions Representation of functions Linear functions Sequences Functions as transformations The image: the range of values of a function Numerical functions are transformations of the line Functions with regularities: one-to-one and onto Compositions of functions The simplest functions The inverse of a function Units conversions and changes of variables The arithmetic operations on functions Solving equations The 2-dimensional space Parametric curves on the plane Functions of two variables Transforming the axes transforms the plane Linear transformations Analytic geometry: the Cartesian system for the Euclidean plane The Euclidean plane: distances The Euclidean plane: angles How complex numbers emerge Classification of quadratic polynomials The complex plane C is the Euclidean space R2 Multidimensional spaces Multiple variables, multiple dimensions Euclidean spaces and Cartesian systems of dimensions 1, 2, 3,... Geometry of distances Where vectors come from Vectors in Rn Algebra of vectors Convex, affine, and linear combinations of vectors The magnitude of a vector Lines as parametric curves The angles between vectors; the dot product Projections and decompositions of vectors Linear operators Where matrices come from Transformations of the plane Linear operators Examining and building linear operators The determinant of a matrix It's a stretch: eigenvalues and eigenvectors The significance of eigenvectors Bases Classification of linear operators according to their eigenvalues Algebra of linear operators and matrices Compositions of linear operators A bird's-eye view of basic calculus What is calculus about? Spaces of functions The sequence of differences The sequence of sums Sums of differences and differences of sums Limits as linear operators Differentiation as a linear operator Integration as a linear operator The Fundamental Theorem of Calculus Linear algebra of power series Exercises Exercises: Background Exercises: Sets and logic Exercises: Coordinate system Exercises: Relations and functions Exercises: Graphs Exercises: Compositions Exercises: Linear equations Exercises: Vector algebra Exercises: Eigenvalues and eigenvectors Exercises: Transformations Exercises: Advanced Index