1 Linear Equations 1
1.1 Linear Equations and Their Solution
1.2 Matrices and Echelon Forms
1.3 How to Use it: Applications of Linear Systems

2 The Vector Space Rn
2.1 Introduction to Vectors: Linear Geometry
2.2 Vectors and the Space Rn
2.3 The Span of a Sequence of Vectors
2.4 Linear independence in Rn
2.5 Subspaces and Bases of Rn
2.6 The Dot Product in Rn

3 Matrix Algebra
3.1 Introduction to Linear Transformations and Matrix Multiplication
3.2 The Product of a Matrix and a Vector
3.3 Matrix Addition and Multiplication
3.4 Invertible Matrices
3.5 Elementary Matrices
3.6 The LU Factorization
3.7 How to Use It: Applications of Matrix Multiplication

4 Determinants
4.1 Introduction to Determinants
4.2 Properties of Determinants
4.3 The Adjoint of a Matrix and Cramer’s Rule

5 Abstract Vector Spaces
5.1 Introduction to Abstract Vector Spaces
5.2 Span and Independence in Vector Spaces
5.3 Dimension of a finite generated vector space
5.4 Coordinate vectors and change of basis
5.5 Rank and Nullity of a Matrix
5.6 Complex Vector Spaces
5.7 Vector Spaces Over Finite Fields
5.8 How to Use it: Error Correcting Codes

6 Linear Transformations
6.1 Introduction to Linear Transformations on Abstract Vector Spaces
6.2 Range and Kernel of a Linear Transformation
6.3 Matrix of a Linear Transformation

7 Eigenvalues and Eigenvectors
7.1 Introduction to Eigenvalues and Eigenvectors
7.2 Diagonalization of Matrices
7.3 Complex Eigenvalues of Real Matrices
7.4 How to Use It: Applications of Eigenvalues and Eigenvectors

8 Orthogonality in Rn
8.1 Orthogonal and Orthonormal Sets in Rn
8.2 The Gram-Schmidt Process and QR-Factorization
8.3 Orthogonal Complements and Projections
8.4 Diagonalization of Real Symmetric Matrices
8.5 Quadratic Forms, Conic Sections and Quadratic Surfaces
8.6 How to Use It: Least Squares Approximation