Chapter 1……. LINEAR SYSTEMS
1.1 Systems of Linear Equations
1.2 Row Reduction & Gauss Jordan Elimination
1.3 Vector Equation
1.4 Matrix Equation
1.5 Matrix Multiplication by a Vector
1.6 Solutions to Linear Systems
1.7 Linear Independence
1.8 Linear Transformations
The Matrix of a Linear Transformation
Chapter 1 Exercises

Chapter 2……. MATRIX OPERATIONS
2.1 Transpose of a Matrix
2.2 Matrix Multiplication
2.3 Properties of Matrix Multiplication
2.4 Identity Matrix
2.5 Matrix Inverses
2.5.1 Inverses of 2x2 Matrices
2.5.2 Inverses of N X N Matrices
2.5.3 Applying Inverse Matrices to Solving Systems of Equations
2.5.4 Properties of Inverse Matrices
2.6 Characterizations of Invertible Matrices
Chapter 2 Exercises

Chapter 3……. VECTOR SPACES
3.1 Vector Spaces and Subspaces
3.2 Column and Null Spaces of a Matrix
3.3 Dimension and Bases
3.4 Coordinates
3.5 Change of Basis
3.6 Vector Coordinates and Bases – Exercises
Chapter 3 Exercises

Chapter 4……. DETERMINANTS
4.1 Introduction to Determinants
4.2 Properties of Determinants
4.3 Cramer’s Rule
4.4 Application of the Determinant

Chapter 5……. EIGENVALUES & DIAGONALIZATION
5.1 Eigenvalues
5.2 The Characteristic Equation
5.3 Diagonalization
5.4 Eigenvalues and Eigenvectors – Exercises
Chapter 5 Exercises

Chapter 6……. INNER PRODUCT SPACES and ORTHOGONALITY
6.1 The Inner Product
6.1.1 Properties of The Inner Product
6.1.2 Interpretation of The Inner Product
6.1.3 Projections
6.2 Orthogonal Complements – Vector Spaces
6.3 Orthogonal Bases
6.4 The Gram-Schmidt Process
6.5 QR Factorization
6.6 Spectral Decomposition of a Symmetric Matrix
6.7 Orthogonality – Exercises
Chapter 6 Exercises