Mathematics for Machine Learning and Data Science 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 Gaussian elimination 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 Principal Component Analysis (PCA)

Week 1: Functions of one variable: Derivative and optimization Lesson 1: Derivatives Example to motivate derivatives: Speedometer Derivative of common functions (c, x, x^2, 1/x) Meaning of e and the derivative of e^x Derivative of log x Existence of derivatives Properties of derivative Lesson 2: Optimization with derivatives Video 1: Intro to optimization: Temperature example Video 2: Optimizing cost functions in ML: Squared loss Video 3: Optimizing cost functions in ML: Log loss Week 2: Functions of two or more variables: Gradients and gradient descent Lesson 1: Gradients and optimization Intro to gradients Example to motivate gradients: Temperature Gradient notation Optimization using slope method: Linear regression Lesson 2: Gradient Descent Optimization using gradient descent: 1 variable Optimization using gradient descent: 2 variable Gradient descent for linear regression Week 3: Optimization in Neural Networks and Newton’s method Lesson 1: Optimization in Neural Networks Perceptron with no activation and squared loss (linear regression) Perceptron with sigmoid activation and log loss (classification) Two-layer neural network with sigmoid activation and log loss Mathematics of Backpropagation Lesson 2: Beyond Gradient Descent: Newton’s Method Root finding with Newton’s method Adapting Newton’s method for optimization Second derivatives and Hessians Multivariate Newton’s method

Week 1: 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 2: 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 3: 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 4: 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