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Course definition: Systems of linear equations, matrices, vector spaces, inner products, linear transformations, determinants, and eigenvalues. (from here)
Chapter 1
Section 1-Systems of Linear Equations
Section 2-Row Reduction and Echelon Forms
Section 3-Vector Equations
Section 4-The Matrix Equation $Ax=\textbf{b}$
Section 5-Solution Sets of Linear Systems
Section 6-Applications of Linear Systems
Section 7-Linear Independence
Section 8-Introduction to Linear Transformations
Section 9-The Matrix of a Linear Transformation
Section 10-Linear Models in Business, Science, and Engineering
Chapter 2
Section 1-Matrix Operations
Section 2-The Inverse of a Matrix
Section 3-Characterizations of Invertible Matrices
Section 4-Partitioned Matrices
Section 5-Matrix Factorizations
Section 6-The Leontief Input-Output Model
Section 7-Applications to Computer Graphics
Section 8-Subspaces of $\mathbb{R}^n$
Section 9-Dimension and Rank
Chapter 3
Section 1-Introduction to Determinants
Section 2-Properties of Determinants
Section 3-Cramer's Rule, Volume, and Linear Transformations
Chapter 4
Section 1-Vector Spaces and Subspaces
Section 2-Null Spaces,Column Spaces, and Linear Transformations
Section 3-Linear Independent Sets; Bases
Section 4-Coordinate Systems
Section 5-The Dimension of a Vector Space
Section 6-Rank
Section 7-Change of Basis
Section 8-Applications to Difference Equations
Section 9-Applications to Markov Chains
Chapter 5
Section 1-Eigenvectors and Eigenvalues
Section 2-The Characteristic Equation
Section 3-Diagonalization
Section 4-Eigenvectors and Linear Transformations
Section 5-Complex Eigenvalues
Section 6-Discrete Dynamical Systems
Section 7-Applications to Differential Equations
Section 8-Iterative Estimates for Eigenvalues
Chapter 6
Section 1-Inner Product, Length, and Orthogonality