These lecture notes are provided for students in MAT232 Linear Algebra and are based on the presentation in Lay. They guide but do not necessarily correspond exactly to the classroom presentation.

• 1.1 Systems of Linear Equations
• 1.2 Row Reduction and Echelon Forms
• 1.3 Vector Equations
• 1.4 The Matrix Equation Ax = b
• 1.5 Solution Sets of Linear Systems
• 1.6 Applications of Linear Systems
• 1.7 Linear Independence
• 1.8 Introduction to Linear Transformations
• 1.9 The Matrix of a Linear Transformation
• 2.1 Matrix Operations
• 2.2 The Inverse of a Matrix
• 2.3 Characterizations of Invertible Matrices
• 2.4 Partitioned Matrices
• 2.5 Matrix Factorizations
• 2.6 The Leontief Input-Output Model
• 2.7 Applications to Computer Graphics
• 3.1 Introduction to Determinants
• 3.2 Properties of Determinants
• 3.3 Cramer's Rule, Volume and Linear Transformations
• 4.1 Vector Spaces and Subspaces
• 4.2 Null & Column Spaces and Linear Transformations
• 4.3 Linearly Independent Sets; Bases
• 4.4 Coordinates Systems
• 4.5 The Dimension of a Vector Space
• 4.6 Rank
• 4.7 Change of Basis
• 4.8 Applications to Difference Equations
• 4.9 Applications to Markov Chains
• 5.1 Eigenvalues and Eigenvectors
• 5.2 The Characteristic Equation
• 5.3 Diagonalization
• 5.4 Eigenvectors and Linear Transformations
• 5.5 Complex Eigenvalues
• 5.6 Discrete Dynamical Systems
• 6.1 Inner Product, Length and Orthogonality
• 6.2 Orthogonal Sets
• 6.3 Orthogonal Projections
• 6.4 The Gram-Schmidt Process
• 6.5 Least-Squares Problems
• 6.7 Inner Product Spaces