1.1 Introduction to Linear Systems
 
1.2 Matrices, Vectors, and Gauss-Jordan Elimination
 
1.3 On the Solutions of Linear Systems; Matrix Algebra
 

2.1 Introduction to Linear Transformations and Their Inverses
 
2.2 Linear Transformations in Geometry
 
2.3 The Inverse of a Linear Transformation
 
2.4 Matrix Product
 

3.1 Image and Kernel of a Linear Transformation
 
3.2 Subspaces of Rn Bases and Linear Independence
 
3.3 The Dimension of a Subspace of Rn
 
3.4 Coordinates
 

4.1 Introduction to Linear Spaces
 
4.2 Linear Transformations and Isomorphisms
 
4.3 The Matrix of a Linear Transformation
 

5.1 Orthogonal Projections and Orthonormal Bases
 
5.2 Gram-Schmidt Process and QR Factorization
 
5.3 Orthogonal Transformations and Orthogonal Matrices
 
5.4 Least Squares and Data Fitting
 
5.5 Inner Product Spaces
 

6.1 Introduction to Determinants
 
6.2 Properties of the Determinant
 
6.3 Geometrical Interpretations of the Determinant; Cramer’s Rule
 

7.1 Dynamical Systems and Eigenvectors: An Introductory Example
 
7.2 Finding the Eigenvalues of a Matrix
 
7.3 Finding the Eigenvectors of a Matrix
 
7.4 Diagonalization
 
7.5 Complex Eigenvalues
 
7.6 Stability
 

8.1 Symmetric Matrices
 
8.2 Quadratic Forms
 
8.3 Singular Values
 

9.1 An Introduction to Continuous Dynamical Systems
 
9.2 The Complex Case: Euler’s Formula
 
9.3 Linear Differential Operators and Linear Differential Equations
 

Visualizing Linear Algebra with Maple
 

Errata