LINEAR ALGEBRA Jim Hefferon Fourth edition

Question

This text covers a standard first course: Gauss's Method, vector spaces, linear maps and matrices, determinants, and eigenvalues and eigenvectors. In addition, each chapter ends with some brief topics, such as applications.

What sets it apart is careful motivation, many examples, and extensive exercise sets. Together, these help each student master the material of the course and also help an instructor develop that student's level of mathematical maturity.

Linear Systems
Solving Linear Systems
Gauss's Method
Describing the Solution Set
General=Particular+Homogeneous
Linear Geometry
Vectors in Space
Length and Angle Measures
Reduced Echelon Form
Gauss-Jordan Reduction
The Linear Combination Lemma
Topic: Computer Algebra Systems
Topic: Input-Output Analysis
Topic: Accuracy of Computations
Topic: Analyzing Networks
Vector Spaces
Definition of Vector Space
Definition and Examples
Subspaces and Spanning Sets
Linear Independence
Definition and Examples
Basis and Dimension
Basis
Dimension
Vector Spaces and Linear Systems
Combining Subspaces
Topic: Fields
Topic: Crystals
Topic: Voting Paradoxes
Topic: Dimensional Analysis
Maps Between Spaces
Isomorphisms
Definition and Examples
Dimension Characterizes Isomorphism
Homomorphisms
Definition
Range Space and Null Space
Computing Linear Maps
Representing Linear Maps with Matrices
Any Matrix Represents a Linear Map
Matrix Operations
Sums and Scalar Products
Matrix Multiplication
Mechanics of Matrix Multiplication
Inverses
Change of Basis
Changing Representations of Vectors
Changing Map Representations
Projection
Orthogonal Projection Into a Line
Gram-Schmidt Orthogonalization
Projection Into a Subspace
Topic: Line of Best Fit
Topic: Geometry of Linear Maps
Topic: Magic Squares
Topic: Markov Chains
Topic: Orthonormal Matrices
Determinants
Definition
Exploration
Properties of Determinants
The Permutation Expansion
Determinants Exist
Geometry of Determinants
Determinants as Size Functions
Laplace's Formula
Laplace's Expansion
Topic: Cramer's Rule
Topic: Speed of Calculating Determinants
Topic: Chiò's Method
Topic: Projective Geometry
Topic: Computer Graphics
Similarity
Complex Vector Spaces
Polynomial Factoring and Complex Numbers
Complex Representations
Similarity
Definition and Examples
Diagonalizability
Eigenvalues and Eigenvectors
Nilpotence
Self-Composition
Strings
Jordan Form
Polynomials of Maps and Matrices
Jordan Canonical Form
Topic: Method of Powers
Topic: Stable Populations
Topic: Page Ranking
Topic: Linear Recurrences
Topic: Coupled Oscillators
Appendix
Statements
Quantifiers
Techniques of Proof
Sets, Functions, and Relations

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