Our Mathematics for Machine Learning course provides a comprehensive foundation of the essential mathematical tools required to study machine learning.
This course is divided into three main categories: linear algebra, multivariable calculus, and probability & statistics. The linear algebra section covers crucial machine learning fundamentals such as matrices, vector spaces, diagonalization, projections, singular value decomposition, and regression. The multivariable calculus section examines vector-valued functions, partial derivatives, and multiple integrals. Finally, the probability and statistics section covers random variables, point estimation, maximum likelihood, hypothesis testing, and confidence intervals.
On completing this course, students will be well-prepared for a university-level machine learning course that tackles concepts such as gradient descent, neural networks, backpropagation, support vector machines, naive Bayes classifiers, and Gaussian mixture models.
After briefly looking at some essential set theory, logic, and vector geometry, students explore matrices in-depth. They will study Gaussian elimination, solve systems of equations, learn about determinants and their properties, and compute inverse matrices.
As part of this course, students perform a deep dive into vector spaces, exploring linear independence, subspaces, bases, dimension, rank, and nullity. Students will generalize key concepts to abstract vector spaces and inner product spaces. Various aspects of orthogonality in vector spaces are considered, including orthogonal sets, complements, orthogonal matrices, orthogonal projections, and the Gram-Schmidt process.
Students will learn how to find the eigenvectors of a matrix, compute a matrix diagonalization, and extend this understanding to symmetric matrices.
In addition, this course discusses various linear algebra applications relevant to machine learning, such as singular value decomposition, linear least-squares, regression, and principal component analysis.
A solid grasp of some key multivariable calculus concepts is needed to understand fundamental machine learning algorithms successfully. In this course, students will become well-versed in partial derivatives and gradient vectors (for gradient descent), the multivariable chain rule (essential for backpropagation), vector-valued functions, and generally, the differential calculus of maps between multi-dimensional vector spaces (which show up when machine learning models are represented using matrix notation). Students will also work with standard multivariable surfaces to build intuition for the concept of a loss surface of a machine learning model. The remainder of the multivariable calculus discusses double integrals, a crucial tool for fully grasping continuous probability distributions and related concepts.
On the probability and statistics side, students will unravel discrete and continuous random variables. They will familiarize themselves with probability density functions, random variable transformations, expectation, moments, and variance. Some important discrete and continuous probability distributions will be discussed in detail.
Students then extend their knowledge of random variables to include joint, marginal, and conditional probability distributions, sums and products of random variables, conditional expectations, and variances. Special attention will be given to combinations of normally distributed random variables and the bivariate normal distribution.
The statistics part of the course concludes with an in-depth study of parametric inference, exploring point estimation, maximum likelihood estimation, and hypothesis testing. Students will also learn to construct confidence intervals for various parameters, including means, proportions, variances, and regression coefficients.
1.1.1. | Special Sets | |
1.1.2. | Statements and Predicates | |
1.1.3. | Equivalent Sets | |
1.1.4. | The Constructive Definition of a Set | |
1.1.5. | The Conditional Definition of a Set | |
1.1.6. | Describing Sets Using Set-Builder Notation | |
1.1.7. | Describing Planar Regions Using Set-Builder Notation | |
1.1.8. | Subsets |
1.2.1. | The Difference of Sets | |
1.2.2. | Set Complements | |
1.2.3. | The Cartesian Product | |
1.2.4. | Visualizing Cartesian Products | |
1.2.5. | Indexed Sets | |
1.2.6. | Sets and Functions |
1.3.1. | Cardinality of Finite Sets | |
1.3.2. | Infinite Sets | |
1.3.3. | Interior and Boundary Points | |
1.3.4. | Interiors and Boundaries of Sets | |
1.3.5. | Open and Closed Sets |
1.4.1. | The Vector Equation of a Line | |
1.4.2. | The Parametric Equations of a Line | |
1.4.3. | The Cartesian Equation of a Line | |
1.4.4. | The Vector Equation of a Plane | |
1.4.5. | The Cartesian Equation of a Plane | |
1.4.6. | The Parametric Equations of a Plane | |
1.4.7. | The Intersection of Two Planes |
1.5.1. | The Hyperbolic Functions | |
1.5.2. | Graphs of the Hyperbolic Functions |
2.6.1. | The Determinant of an NxN Matrix | |
2.6.2. | Finding Determinants Using Laplace Expansions | |
2.6.3. | Basic Properties of Determinants | |
2.6.4. | Further Properties of Determinants | |
2.6.5. | Row and Column Operations on Determinants | |
2.6.6. | Conditions When a Determinant Equals Zero |
2.7.1. | Systems of Equations as Augmented Matrices | |
2.7.2. | Row Echelon Form | |
2.7.3. | Solving Systems of Equations Using Back Substitution | |
2.7.4. | Elementary Row Operations | |
2.7.5. | Creating Rows or Columns Containing Zeros Using Gaussian Elimination | |
2.7.6. | Solving 2x2 Systems of Equations Using Gaussian Elimination | |
2.7.7. | Solving 2x2 Singular Systems of Equations Using Gaussian Elimination | |
2.7.8. | Solving 3x3 Systems of Equations Using Gaussian Elimination | |
2.7.9. | Identifying the Pivot Columns of a Matrix | |
2.7.10. | Solving 3x3 Singular Systems of Equations Using Gaussian Elimination | |
2.7.11. | Reduced Row Echelon Form | |
2.7.12. | Gaussian Elimination For NxM Systems of Equations |
2.8.1. | Finding the Inverse of a 2x2 Matrix Using Row Operations | |
2.8.2. | Finding the Inverse of a 3x3 Matrix Using Row Operations | |
2.8.3. | Matrices With Easy-to-Find Inverses | |
2.8.4. | The Invertible Matrix Theorem in Terms of 2x2 Systems of Equations | |
2.8.5. | Triangular Matrices |
2.9.1. | Affine Transformations | |
2.9.2. | The Image of an Affine Transformation | |
2.9.3. | The Inverse of an Affine Transformation |
3.10.1. | Vectors in N-Dimensional Euclidean Space | |
3.10.2. | Linear Combinations of Vectors in N-Dimensional Euclidean Space | |
3.10.3. | Linear Span of Vectors in N-Dimensional Euclidean Space | |
3.10.4. | Linear Dependence and Independence |
3.11.1. | Subspaces of N-Dimensional Space | |
3.11.2. | Subspaces of N-Dimensional Space: Geometric Interpretation | |
3.11.3. | The Column Space of a Matrix | |
3.11.4. | The Null Space of a Matrix |
3.12.1. | Finding a Basis of a Span | |
3.12.2. | Finding a Basis of the Column Space of a Matrix | |
3.12.3. | Finding a Basis of the Null Space of a Matrix | |
3.12.4. | Expressing the Coordinates of a Vector in a Given Basis | |
3.12.5. | Writing Vectors in Different Bases | |
3.12.6. | The Change-of-Coordinates Matrix | |
3.12.7. | Changing a Basis Using the Change-of-Coordinates Matrix |
3.13.1. | The Dimension of a Span | |
3.13.2. | The Rank of a Matrix | |
3.13.3. | The Dimension of the Null Space of a Matrix | |
3.13.4. | The Invertible Matrix Theorem in Terms of Dimension, Rank and Nullity | |
3.13.5. | The Rank-Nullity Theorem |
4.14.1. | The Eigenvalues and Eigenvectors of a 2x2 Matrix | |
4.14.2. | Calculating the Eigenvalues of a 2x2 Matrix | |
4.14.3. | Calculating the Eigenvectors of a 2x2 Matrix | |
4.14.4. | The Characteristic Equation of a Matrix | |
4.14.5. | Calculating the Eigenvectors of a 3x3 Matrix With Distinct Eigenvalues | |
4.14.6. | Calculating the Eigenvectors of a 3x3 Matrix in the General Case |
4.15.1. | Diagonalizing a 2x2 Matrix | |
4.15.2. | Diagonalizing a 3x3 Matrix With Distinct Eigenvalues | |
4.15.3. | Diagonalizing a 3x3 Matrix in the General Case | |
4.15.4. | Symmetric Matrices | |
4.15.5. | Diagonalization of 2x2 Symmetric Matrices | |
4.15.6. | Diagonalization of 3x3 Symmetric Matrices |
5.16.1. | The Dot Product in N-Dimensional Euclidean Space | |
5.16.2. | The Norm of a Vector in N-Dimensional Euclidean Space | |
5.16.3. | Introduction to Abstract Vector Spaces | |
5.16.4. | Defining Abstract Vector Spaces | |
5.16.5. | Inner Product Spaces |
5.17.1. | Orthogonal Vectors in Euclidean Spaces | |
5.17.2. | The Cauchy-Schwarz Inequality and the Angle Between Two Vectors | |
5.17.3. | Orthogonal Complements | |
5.17.4. | Orthogonal Sets in Euclidean Spaces | |
5.17.5. | Orthogonal Matrices | |
5.17.6. | Orthogonal Linear Transformations |
5.18.1. | Projecting Vectors Onto One-Dimensional Subspaces | |
5.18.2. | The Components of a Vector with Respect to an Orthogonal or Orthonormal Basis | |
5.18.3. | Projecting Vectors Onto Subspaces in Euclidean Spaces (Orthogonal Bases) | |
5.18.4. | Projecting Vectors Onto Subspaces in Euclidean Spaces (Arbitrary Bases) | |
5.18.5. | Projecting Vectors Onto Subspaces in Euclidean Spaces (Arbitrary Bases): Applications | |
5.18.6. | The Gram-Schmidt Process for Two Vectors |
6.19.1. | Bilinear Forms | |
6.19.2. | Quadratic Forms | |
6.19.3. | Change of Variables in Quadratic Forms | |
6.19.4. | Positive-Definite and Negative-Definite Quadratic Forms | |
6.19.5. | Constrained Optimization of Quadratic Forms | |
6.19.6. | Constrained Optimization of Quadratic Forms: Determining Where Extrema are Attained |
6.20.1. | The Singular Values of a Matrix | |
6.20.2. | Computing the Singular Values of a Matrix | |
6.20.3. | Singular Value Decomposition of 2x2 Matrices | |
6.20.4. | Singular Value Decomposition of 2x2 Matrices With Zero or Repeated Eigenvalues | |
6.20.5. | Singular Value Decomposition of Larger Matrices | |
6.20.6. | Singular Value Decomposition and the Pseudoinverse Matrix |
7.21.1. | Introduction to Principal Component Analysis | |
7.21.2. | Computing Principal Components | |
7.21.3. | The Connection Between PCA and SVD |
7.22.1. | The Least-Squares Solution of a Linear System (Without Collinearity) | |
7.22.2. | The Least-Squares Solution of a Linear System (With Collinearity) |
7.23.1. | Linear Regression | |
7.23.2. | Polynomial Regression | |
7.23.3. | Multiple Linear Regression |
8.24.1. | Ellipsoids | |
8.24.2. | Hyperboloids | |
8.24.3. | Paraboloids | |
8.24.4. | Elliptic Cones | |
8.24.5. | Cylinders | |
8.24.6. | Identifying Quadric Surfaces |
8.25.1. | The Domain of a Multivariable Function | |
8.25.2. | Level Curves | |
8.25.3. | Limits and Continuity of Multivariable Functions | |
8.25.4. | Introduction to Partial Derivatives | |
8.25.5. | Computing Partial Derivatives Using the Rules of Differentiation | |
8.25.6. | Geometric Interpretations of Partial Derivatives | |
8.25.7. | Partial Differentiability of Multivariable Functions | |
8.25.8. | Higher-Order Partial Derivatives | |
8.25.9. | Equality of Mixed Partial Derivatives | |
8.25.10. | Tangent Planes to Surfaces | |
8.25.11. | Linearization of Multivariable Functions | |
8.25.12. | The Multivariable Chain Rule |
8.26.1. | The Domain of a Vector-Valued Function | |
8.26.2. | Tangent Vectors and Tangent Lines to Curves | |
8.26.3. | The Gradient Vector | |
8.26.4. | Directional Derivatives | |
8.26.5. | The Multivariable Chain Rule in Vector Form |
8.27.1. | The Jacobian | |
8.27.2. | The Inverse Function Theorem | |
8.27.3. | The Jacobian of a Three-Dimensional Transformation | |
8.27.4. | The Derivative of a Multivariable Function | |
8.27.5. | The Second Derivative of a Multivariable Function | |
8.27.6. | Second-Degree Taylor Polynomials of Multivariable Functions |
8.28.1. | Partitions of Intervals | |
8.28.2. | Calculating Double Summations Over Partitions | |
8.28.3. | Approximating Volumes Using Lower Riemann Sums | |
8.28.4. | Approximating Volumes Using Upper Riemann Sums | |
8.28.5. | Lower Riemann Sums Over General Rectangular Partitions | |
8.28.6. | Upper Riemann Sums Over General Rectangular Partitions | |
8.28.7. | Defining Double Integrals Using Lower and Upper Riemann Sums |
8.29.1. | Double Integrals Over Rectangular Domains | |
8.29.2. | Double Integrals Over Non-Rectangular Domains | |
8.29.3. | Properties of Double Integrals | |
8.29.4. | Type I and II Regions in Two-Dimensional Space | |
8.29.5. | Double Integrals Over Type I Regions | |
8.29.6. | Double Integrals Over Type II Regions |
9.30.1. | The Law of Total Probability (Extended) | |
9.30.2. | Bayes' Theorem | |
9.30.3. | Extending Bayes' Theorem |
9.31.1. | Probability Density Functions of Continuous Random Variables | |
9.31.2. | Calculating Probabilities With Continuous Random Variables | |
9.31.3. | Continuous Random Variables Over Infinite Domains | |
9.31.4. | Cumulative Distribution Functions for Continuous Random Variables | |
9.31.5. | Approximating Discrete Random Variables as Continuous | |
9.31.6. | Simulating Random Observations |
9.32.1. | One-to-One Transformations of Discrete Random Variables | |
9.32.2. | Many-to-One Transformations of Discrete Random Variables | |
9.32.3. | The Distribution Function Method | |
9.32.4. | The Change-of-Variables Method for Continuous Random Variables | |
9.32.5. | The Distribution Function Method With Many-to-One Transformations |
9.33.1. | Expected Values of Discrete Random Variables | |
9.33.2. | Properties of Expectation for Discrete Random Variables | |
9.33.3. | Moments of Discrete Random Variables | |
9.33.4. | Variance of Discrete Random Variables | |
9.33.5. | Properties of Variance for Discrete Random Variables | |
9.33.6. | Expected Values of Continuous Random Variables | |
9.33.7. | Moments of Continuous Random Variables | |
9.33.8. | Variance of Continuous Random Variables | |
9.33.9. | The Rule of the Lazy Statistician |
9.34.1. | The Bernoulli Distribution | |
9.34.2. | Mean and Variance of the Binomial Distribution | |
9.34.3. | The Discrete Uniform Distribution | |
9.34.4. | Modeling With Discrete Uniform Distributions | |
9.34.5. | Mean and Variance of Discrete Uniform Distributions | |
9.34.6. | The Poisson Distribution | |
9.34.7. | Modeling With the Poisson Distribution |
9.35.1. | The Continuous Uniform Distribution | |
9.35.2. | Mean and Variance of Continuous Uniform Distributions | |
9.35.3. | Modeling With Continuous Uniform Distributions | |
9.35.4. | The Gamma Function | |
9.35.5. | The Chi-Square Distribution | |
9.35.6. | The Student's T-Distribution | |
9.35.7. | The Exponential Distribution |
10.36.1. | Double Summations | |
10.36.2. | Joint Distributions for Discrete Random Variables | |
10.36.3. | Marginal Distributions for Discrete Random Variables | |
10.36.4. | Independence of Discrete Random Variables | |
10.36.5. | Conditional Distributions for Discrete Random Variables | |
10.36.6. | The Joint CDF of Two Discrete Random Variables |
10.37.1. | Joint Distributions for Continuous Random Variables | |
10.37.2. | Marginal Distributions for Continuous Random Variables | |
10.37.3. | Independence of Continuous Random Variables | |
10.37.4. | Conditional Distributions for Continuous Random Variables | |
10.37.5. | The Joint CDF of Two Continuous Random Variables | |
10.37.6. | Properties of the Joint CDF of Two Continuous Random Variables |
10.38.1. | Expected Values of Sums and Products of Random Variables | |
10.38.2. | Variance of Sums of Independent Random Variables | |
10.38.3. | Computing Expected Values From Joint Distributions | |
10.38.4. | Conditional Expectation for Discrete Random Variables | |
10.38.5. | Conditional Variance for Discrete Random Variables | |
10.38.6. | Conditional Expectation for Continuous Random Variables | |
10.38.7. | Conditional Variance for Continuous Random Variables | |
10.38.8. | The Rule of the Lazy Statistician for Two Random Variables |
10.39.1. | The Covariance of Two Random Variables | |
10.39.2. | Variance of Sums of Random Variables | |
10.39.3. | The Correlation Coefficient for Two Random Variables | |
10.39.4. | The Covariance Matrix |
10.40.1. | Normal Approximations of Binomial Distributions | |
10.40.2. | Combining Two Normally Distributed Random Variables | |
10.40.3. | Combining Multiple Normally Distributed Random Variables | |
10.40.4. | I.I.D Normal Random Variables | |
10.40.5. | The Bivariate Normal Distribution |
11.41.1. | The Sample Mean | |
11.41.2. | Statistics and Sampling Distributions | |
11.41.3. | Variance of Sample Means | |
11.41.4. | The Sample Variance | |
11.41.5. | Sample Means From Normal Populations | |
11.41.6. | The Central Limit Theorem | |
11.41.7. | Sampling Proportions From Finite Populations | |
11.41.8. | Point Estimates of Population Proportions | |
11.41.9. | The Sample Covariance Matrix |
11.42.1. | Product Notation | |
11.42.2. | Logarithmic Differentiation | |
11.42.3. | Likelihood Functions for Discrete Probability Distributions | |
11.42.4. | Log-Likelihood Functions for Discrete Probability Distributions | |
11.42.5. | Likelihood Functions for Continuous Probability Distributions | |
11.42.6. | Log-Likelihood Functions for Continuous Probability Distributions | |
11.42.7. | Maximum Likelihood Estimation |
11.43.1. | One-Tailed Hypothesis Tests | |
11.43.2. | Two-Tailed Hypothesis Tests | |
11.43.3. | Type I and Type II Errors in Hypothesis Testing | |
11.43.4. | Hypothesis Tests for One Mean: Known Population Variance | |
11.43.5. | Hypothesis Tests for One Mean: Unknown Population Variance | |
11.43.6. | Hypothesis Tests for Two Means: Known Population Variances |
11.44.1. | Confidence Intervals for One Mean: Known Population Variance | |
11.44.2. | Confidence Intervals for One Mean: Unknown Population Variance | |
11.44.3. | Confidence Intervals for Proportions | |
11.44.4. | Confidence Intervals for Two Means: Known and Unequal Population Variances | |
11.44.5. | Confidence Intervals for Variances | |
11.44.6. | Confidence Intervals for Slope Parameters in Linear Regression | |
11.44.7. | Confidence Intervals for Intercept Parameters in Linear Regression |