Linear Algebra:

Linear Equations

• Systems of Linear Equations
• Row Reduction and Echelon Forms
• Vector Equations
• The Matrix Equation Ax=b
• Solution Sets of Linear Systems
• Linear Independence
• Linear Transformations and the Matrix of a Linear Transformation
• Applications of Linear Systems and Linear Models in Business, Science and Engineering

Matrix Algebra

• Matrix Operations and Determinants
• The Inverse of a Matrix
• Characterizations of Invertible Matrices
• Partitioned Matrices
• Matrix Factorization (LU Decomposition)
• Applications to Computer Graphics

Determinants

•  Properties to Determinants
• Cramer’s Rule, Volume, and Linear Transformations

Vector Spaces 

• Vector Spaces and Subspaces
• Null Spaces, Column Spaces, Row Spaces and Linear Transformations
• Linearly Independent Sets; Bases
• Coordinate Systems
• The Dimension of a Vector Space
• Rank
• Change of Basis
• Applications to Markov Chains

Eigenvalues and Eigenvectors

• Eigenvectors and Eigenvalues
• The Characteristic Equation
• Diagonolization
• Eigenvectors and Linear Transformations
• Complex Eigenvalues

Orthogonality and Least Squares

•  Inner Product, Length, and Orthogonality
• Orthogonal Sets
• Orthogonal Projections
• The Gram-Schmidt Process and QR Factorization
• Least-Squares Problems
• Applications to Linear Models (Line of Regressions)
• Inner Product Spaces

Symmetric Matrices and Quadratic Forms

• Diagonalization of Symmetric Matrices
• Quadratic Forms

Complex Numbers: