Mathematics for ML & DS Specialization

Week 1: Systems of Linear Equations

Lesson 1: Systems of Linear equations: two variables

• Machine learning motivation
• Systems of sentences
• Systems of equations
• Systems of equations as lines
• A geometric notion of singularity
• Singular vs nonsingular matrices
• Linear dependence and independence
• The determinant

Lesson 2: Systems of Linear Equations: three variables

• Systems of equations (3×3)
• Singular vs non-singular (3×3)
• Systems of equations as planes (3×3)
• Linear dependence and independence (3×3)
• The determinant (3×3)

Week 2: Solving systems of Linear Equations

Lesson 1: Solving systems of Linear Equations: Elimination

• Machine learning motivation
• Solving non-singular systems of linear equations
• Solving singular systems of linear equations
• Solving systems of equations with more variables
• Matrix row-reduction
• Row operations that preserve singularity

Lesson 2: Solving systems of Linear Equations: Row Echelon Form and Rank

• The rank of a matrix
• The rank of a matrix in general
• Row echelon form
• Row echelon form in general
• Reduced row echelon form

Week 3: Vectors and Linear Transformations

Lesson 1: Vectors

• Norm of a vector
• Sum and difference of vectors
• Distance between vectors
• Multiplying a vector by a scalar
• The dot product
• Geometric Dot Product
• Multiplying a matrix by a vector
• Lab: Vector Operations: Scalar Multiplication, Sum and Dot Product of Vectors

Lesson 2: Linear transformations

• Matrices as linear transformations
• Linear transformations as matrices
• Matrix multiplication
• The identity matrix
• Matrix inverse
• Which matrices have an inverse?
• Neural networks and matrices

Week 4: Determinants and Eigenvectors

Lesson 1: Determinants In-depth

• Machine Learning Motivation
• Singularity and rank of linear transformation
• Determinant as an area
• Determinant of a product
• Determinants of inverses

Lesson 2: Eigenvalues and Eigenvectors

• Bases in Linear Algebra
• Span in Linear Algebra
• Interactive visualization: Linear Span
• Eigenbases
• Eigenvalues and eigenvectors

Week 7: Introduction to probability and random variables

Lesson 1: Introduction to probability

• Concept of probability: repeated random trials
• Conditional probability and independence 
• Discriminative learning and conditional probability
• Bayes theorem 

Lesson 2: Random variables

• Random variables
• Cumulative distribution function
• Discrete random variables: Bernoulli distribution
• Discrete random variables: Binomial distribution
• Probability mass function
• Continuous random variables: Uniform distribution 
• Continuous random variables: Gaussian distribution
• Continuous random variables: Chi squared distribution
• Probability distribution function

Week 8-9: Describing distributions and random vectors

Lesson 1: Describing distributions

• Measures of central tendency: mean, median, mode
• Expected values
• Quantiles and box-plots 
• Measures of dispersion: variance, standard deviation

Lesson 2: Random vectors

• Joint distributions
• Marginal and conditional distributions
• Independence
• Measures of relatedness: covariance
• Multivariate normal distribution

Week 10-11: Introduction to statistics

Lesson 1: Sampling and point estimates

• Population vs. sample 
• Describing samples: sample proportion and sample mean
• Distribution of sample mean and proportion: Central Limit Theorem 
• Point estimates
• Biased vs Unbiased estimates 

Lesson 2: Maximum likelihood estimation

• ML motivation example: Linear Discriminant Analysis
• Likelihood
• Intuition behind maximum likelihood estimation
• MLE: How to get the maximum using calculus

Lesson 3: Bayesian statistics 

• ML motivation example: Naive Bayes
• Frequentist vs. Bayesian statistics
• A priori/ a posteriori distributions
• Bayesian estimators: posterior mean, posterior median, MAP

Week 12: Interval statistics and Hypothesis testing

Lesson 1: Confidence intervals

• Margin of error
• Interval estimation
• Confidence Interval for mean of population
• CI for parameters in linear regression
• Prediction Interval

Lesson 2: Hypothesis testing

• ML Motivation: AB Testing
• Criminal trial
• Two types of errors
• Test for proportion and means
• Two sample inference for difference between groups 
• ANOVA
• Power of a test