Linear Algebra and Probability for Computer Science Applications

MATLAB
Desk calculator operations
Booleans
Nonstandard numbers
Loops and conditionals
Script file
Functions
Variable scope and parameter passing

I: Linear Algebra
Vectors
Definition of vectors
Applications of vectors
Basic operations on vectors
Dot product
Vectors in MATLAB: Basic operations
Plotting vectors in MATLAB
Vectors in other programming languages

Matrices
Definition of matrices
Applications of matrices
Simple operations on matrices
Multiplying a matrix times a vector
Linear transformation
Systems of linear equations
Matrix multiplication
Vectors as matrices
Algebraic properties of matrix multiplication
Matrices in MATLAB

Vector Spaces
Subspaces
Coordinates, bases, linear independence
Orthogonal and orthonormal basis
Operations on vector spaces
Null space, image space, and rank
Systems of linear equations
Inverses
Null space and Rank in MATLAB
Vector spaces
Linear independence and bases
Sum of vector spaces
Orthogonality
Functions
Linear transformations
Inverses
Systems of linear equations
The general definition of vector spaces

Algorithms
Gaussian elimination: Examples
Gaussian elimination: Discussion
Computing a matrix inverse
Inverse and systems of equations in MATLAB
Ill-conditioned matrices
Computational complexity

Geometry
Arrows
Coordinate systems
Simple geometric calculations
Geometric transformations

Change of Basis, DFT, and SVD
Change of coordinate system
The formula for basis change
Confusion and how to avoid it
Nongeometric change of basis
Color graphics
Discrete Fourier transform (Optional)
Singular value decomposition
Further properties of the SVD
Applications of the SVD
MATLAB

II: Probability
Probability
The interpretations of probability theory
Finite sample spaces
Basic combinatorial formulas
The axioms of probability theory
Conditional probability
The likelihood interpretation
Relation between likelihood and sample space probability
Bayes’ law
Independence
Random variables
Application: Naive Bayes’ classification

Numerical Random Variables
Marginal distribution
Expected value
Decision theory
Variance and standard deviation
Random variables over infinite sets of integers
Three important discrete distributions
Continuous random variables
Two important continuous distributions
MATLAB

Markov Models
Stationary probability distribution
PageRank and link analysis
Hidden Markov models and the k-gram model

Confidence Intervals
The basic formula for confidence intervals
Application: Evaluating a classifier
Bayesian statistical inference (Optional)
Confidence intervals in the frequentist viewpoint: (Optional)
Hypothesis testing and statistical significance
Statistical inference and ESP

Monte Carlo Methods
Finding area
Generating distributions
Counting
Counting solutions to DNF (Optional)
Sums, expected values, integrals
Probabilistic problems
Resampling
Pseudo-random numbers
Other probabilistic algorithms
MATLAB

Information and Entropy
Information
Entropy
Conditional entropy and mutual information
Coding
Entropy of numeric and continuous random variables
The principle of maximum entropy
Statistical inference

Maximum Likelihood Estimation
Sampling
Uniform distribution
Gaussian distribution: Known variance
Gaussian distribution: Unknown variance
Least squares estimates
Principal component analysis
Applications of PCA

Biography

Ernest Davis is a computer science professor in the Courant Institute of Mathematical Sciences at New York University. He earned a Ph.D. in computer science from Yale University. Dr. Davis is a member of the American Association of Artificial Intelligence and is a reviewer for many journals. His research primarily focuses on spatial and physical reasoning.