MAT2253
Applied Linear Algebra
Prerequisite: MAT1214/MAT1213 Calculus I
This comprehensive course in linear algebra provides an in-depth exploration of core concepts and their applications to optimization, data analysis, and neural networks. Students will gain a strong foundation in the fundamental notions of linear systems of equations, vectors, and matrices, as well as advanced topics such as eigenvalues, eigenvectors, and canonical solutions to linear systems of differential equations. The course also explores he critical techniques of calculus operations in vectors and matrices, optimization, and Taylor series in one and multiple variables. By the end of the course, students will have a thorough understanding of the mathematical framework underlying principal component analysis, gradient descent, and the implementation of simple neural networks.
The primary textbook is "Mathematics for Machine Learning" by Deisenroth, Faisal, and Ong, 2020, Cambridge University Press. The book is available for free for personal use at https://mml-book.github.io/book/mml-book.pdf
The secondary textbook is "Pattern Recognition and Machine Learning" by Bishop, 2006, Springer Information Science and Statistics. The book is available for free for personal use at https://www.microsoft.com/en-us/research/uploads/prod/2006/01/Bishop-Pattern-Recognition-and-Machine-Learning-2006.pdf
| Session | Section | Topic | Prerequisites | SLOs |
|---|---|---|---|---|
| 1 | 2.1 | Systems of Linear Equations | ||
| 2 | 2.2 | Matrices | ||
| 3 | 2.3 | Solving systems of linear equations | ||
| 4 | 3.1, 3.2, 3.3 | Norms, Inner Products, Lengths & Distances | ||
| 5 | 3.4 | Angles & orthogonality | ||
| 6 | 2.4, 2.5 | Vector spaces & Linear Independendence | ||
| 7 | Mini-test | |||
| 8 | 2.6 | Basis & Rank | ||
| 9 | 2.7 | Linear Mappings | ||
| 10 | 4.1 | Determinant and Traces | ||
| 11 | 4.2 | Eigenvalues & Eigenvectors | ||
| 12 | 4.3, 4.4 | Matrix Factorization | ||
| 13 | 3.5 | Orthonormal Basis | ||
| 14 | 3.7 | Inner Product of Functions | ||
| 15 | 3.9 | Rotations | ||
| 16 | Mini-test | |||
| 17 | 5.1 | Vector Calculus | ||
| 18 | 5.1 | Taylor Series | ||
| 19 | 5.1 | Differentiation Rules Review | ||
| 20 | 5.2 | Partial Derivatives | ||
| 21 | 5.2 | Gradients - Examples, visualizations, computaiton | ||
| 22 | 5.2 | Rules for Partial Differentiation & Chain Rule | ||
| 23 | Mini-test | |||
| 24 | 5.3 | Gradients of Vector-Valued Functions | ||
| 25 | 5.3 | Gradients of Vector-Valued Functions | ||
| 26 | 5.4, Dhrymes 78 | Gradients of Matrices | ||
| 27 | 5.5, Dhrymes 78 | Useful Identities for Computing Gradients | ||
| 28 | 5.7 | Higher-Order Derivatives | ||
| 29 | Notes | Minimization via Newton's Method & Backpropagation | ||
| 30 | Min-test | |||
| 31 | 5.8 | Linearization & Multivariate Taylor Series | ||
| 32 | Notes | Linear optimization: Simplex method | ||
| 33 | 7.1 | Optimization Using Gradient Descent | ||
| 34 | 7.2 | Constrained Optimization and Lagrange Multipliers | ||
| 35 | 7.3 | Convex Optimization | ||
| 36 | Mini-Test | |||
| 37 | Bishop, Duda et al. | Feed-forward Artificial Neural Networks | ||
| 38 | Bishop, Duda et al. | Backpropagation in ANNs | ||
| 39 | Bishop, Duda et al. | Activation Functions: Linear & Nonlinear | ||
| 40 | Bishop, Duda et al. | Step-by-step simple ANN | ||
| 41 | Bishop, Duda et al. | Measures of performance | ||
| 42 | Bishop, Duda et al. | More complex architectures of ANNs | ||
| 43 | Mini-test | |||
| 44 | Final project | |||
| 45 | Review |
