Linear Algebra with Applications

Chapter 1: Linear Equations

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

Chapter 2: Linear Transformations

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

Chapter 3: Subspaces of Rn and Their Dimensions

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

Chapter 4: Linear Spaces

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

Chapter 5: Orthogonality and Least Squares

• • 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

Chapter 6: Determinants

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

Chapter 7: Eigenvalues and Eigenvectors

• • 7.1 Diagonalization
• • 7.2 Finding the Eigenvalues of a Matrix
• • 7.3 Finding the Eigenvectors of a Matrix
• • 7.4 More on Dynamical Systems
• • 7.5 Complex Eigenvalues
• • 7.6 Stability

Chapter 8: Symmetric Matrices and Quadratic Forms

• • 8.1 Symmetric Matrices
• • 8.2 Quadratic Forms
• • 8.3 Singular Values

Chapter 9: Linear Differential Equations

• • 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

Chapter Appendix A: Vectors

Chapter Appendix B: Techniques of Proof