**Chapter Zero The Language of Mathematics: Sets, Axioms, Theorems & Proofs**

**Chapter 1 The Canvas of Linear Algebra: Euclidean Spaces and Subspaces**

Euclidean Spaces

The Span of a Set of Vectors

The Dot Product and Orthogonality

Systems of Linear Equations

Linear Systems and Linear Independence

The Relationship Between Independent Sets and Spanning Sets

Subspaces of Euclidean Spaces

Basis and Dimension

**Chapter 2 Adding Movement and Colors: Linear Transformations on Euclidean Spaces**

Mapping Spaces: Introduction to Linear Transformations

Rotations, Projections and Reflections

Operations on Linear Transformations and Matrices

Properties of Operations on Linear Transformations and Matrices

Kernel, Range, One-to-One and Onto Transformations

Invertible Operators and Matrices

Finding the Inverse of a Matrix

Conditions for Invertibility

Diagonal, Triangular, and Symmetric Matrices

**Chapter 3 From The Real to The Abstract: General Vector Spaces**

The Axioms for a Vector Space

Linearity Properties for Finite Sets of Vectors

Linearity Properties for Infinite Sets of Vectors

Subspaces, Basis and Dimension

Linear Transformations on General Vector Spaces

Coordinate Vectors and Matrices for Linear Transformations

One-to-One and Onto Linear Transformations

Compositions of Linear Transformations

Isomorphisms and their Applications

**Chapter 4 Peeling The Onion: The Subspace Structure of Vector Spaces**

The Join and Intersection of Two Subspaces

Restricting Linear Transformations and the Role of the Rowspace

The Image and Preimage of Subspaces

Cosets and Quotient Spaces

Three Isomorphism Theorems

**Chapter 5 From Square to Scalar: Permutation Theory and Determinants**

Permutations and The Determinant Concept

A General Determinant Formula

Computational Tools and Properties of Determinants

The Adjugate Matrix and Cramer’s Rule

**Chapter 6. Painting the Lines: Eigentheory, Diagonalization and Similarity**

The Eigentheory of Square Matrices

Computational Techniques for Eigentheory

Diagonalization of Square Matrices

Change of Basis and Linear Transformations on Euclidean Spaces

Change of Basis for Abstract Spaces and Determinants for Operators

Similarity and The Eigentheory of Operators

**Chapter 7 Geometry in the Abstract: Inner Product Spaces**

The Axioms for an Inner Product Space

Geometric Constructions in Inner Product Spaces

Orthonormal Sets and The Gram-Schmidt Algorithm

Orthogonal Complements and Decompositions

Orthonormality and Projection Operators

Orthogonal Matrices

Orthogonal Diagonalization of Symmetric Matrices

The Method of Least Squares

The QR-Decomposition

**Chapter 8 Imagine That: Complex Spaces and The Spectral Theorems**

The Field of Complex Numbers

Complex Vector Spaces

Complex Inner Products

Complex Linear Transformations and The Adjoint

Normal Matrices

The Spectral Theorems

**Chapter 9 The Big Picture: The Fundamental Theorem of Linear Algebra and Applications**

Balancing Chemical Equations

Basic Circuit Analysis

Recurrence Relations

Introduction to Quadratic Forms

Rotations of Conics

Positive Definite Quadratic Forms and Matrices

The Fundamental Theorem of Linear Algebra

The Singular Value Decomposition

Applications of the SVD

**Appendices: **

A. The Real Number System

B. Logical Symbols and Truth Tables

**Selected Answers to the Exercises**

**Glossary of Symbols**

**Subject Index**