Systems of Linear Equations
Row Operations
Echelon Form
Existence and Uniqueness of Solutions
Vector Equations
The Matrix Equation Ax=b
Balancing Chemical Equations
Network Flow
Linear Independence
Linear Transformations
Superposition Principle
Matrix of a Transformation
Geometric Transformations
One-to-one Transformations

Matrix Arithmetic
Row-Column Rule for AB
Matrix Inversion
Inversion and Ax=b
Properties of Inverse
Elementary Matrices
Row Reduction and Inverses
Invertible Matrix Theorem
Partitioned Matrices
Column-Row Expansion for AB
LU Factorization
LU and Electrical Circuits
Leontief Input-Output Model
Basic Computer Graphics

Definition of Determinants
Cofactor Expansion
Properties of Determinants
Row Reducing for Determinants

Definition of a Vector Space
Examples: Polynomials, Arrows, Sequences, Functions, Matrices
Definition of Subspaces
Spanning Sets
Null Space of a Matrix
Column Space of a Matrix
Kernel and Range of Transformations
Basis of a Subspace
Pivot Columns as Basis
Unique Representation Theorem Coordinates from a Basis
Euclidean Coordinates
Dimension of a Vector Space
Basis Theorem
Row Spaces
Rank Theorem
Change of Coordinates
Difference Equations
Markov Chains

Definition of Eigenvectors
Triangular Matrices
Distinct Eigenvalues
Characteristic Equation
Similar Matrices
Diagonalization of Matrices
Eigenvectors and Transformations
Complex Eigenvalues
Predator-Prey Systems

Inner Products
Length of a Vector
Distance between Vectors
Orthogonal Vectors
The Pythagorean Theorem
Orthogonal Complement
Angles and Vectors
Orthogonal Basis
Orthogonal Projection
Orthonormal Sets
Orthogonal Matrices
Orthogonal Decomposition
Geometric View of Projections
Best Approximation Theorem
Gram-Schmidt Process
QR Factorization
Least Squares Problem
Solution to Least Squares Problem
Normal Equations
Inverse of A^TA
QR Factorization and Least Squares
Least-Squares Line of Best Fit
Least-Squares Fitting of Curves

Diagonalization of Symmetric Matrices
Spectral Theorem
Quadratic Form
Change of Variable in a Quadratic Form
Principal Axes
Quadratic Forms and Eigenvalues