Difference between revisions of "MAT2253"

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Prerequisite: [[MAT1214]]/[[MAT1213]] Calculus I
 
Prerequisite: [[MAT1214]]/[[MAT1213]] Calculus I
  
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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 delves into the 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.
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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.
  
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==List of Topics==
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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
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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
  
 
{| class="wikitable"
 
{| class="wikitable"
 
! Week !! Section !! Topic !! Prerequisites !! SLOs
 
! Week !! Section !! Topic !! Prerequisites !! SLOs
 
|-
 
|-
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| 1 ||  || Notions of linear systems of equations to introduce the concepts of vector and matrices.   ||  ||  
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| 1 || 2.1 || Systems of Linear Equations ||  ||  
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|-
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| rowspan="2" | 2 || 2.2 || Matrices ||  ||
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|-
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| 2.3 || Solving systems of linear equations ||  ||
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|-
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| 3 || 2.4 || Vector spaces ||  ||
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|-
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| rowspan="2" | 4 || 2.5 || Linear Independence ||  ||
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|-
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| 2.6 || Basis & Rank ||  ||
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|-
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| rowspan="2" | 5 || Exam 1 ||  ||  ||
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|-
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| 2.7 || Linear Mappings ||  ||
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|-
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| rowspan="2" | 6 || 2.7 || Linear Mappings (examples) ||  ||
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|-
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| 4.1 || Determinant and Traces ||  ||
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|-
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| rowspan="2" | 7 || 4.2 || Eigenvalues & Eigenvectors ||  ||
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|-
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| 4.3, 4.4 || Matrix Factorizations (Diagonalization) ||  ||
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|-
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| rowspan="2" | 8 || 3.1, 3.2, 3.3 || Norms, Inner Products, Lengths & Distances ||  ||
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|-
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| 3.4 || Angles & orthogonality ||  ||
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|-
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| rowspan="3" | 9 || 3.5 || *Orthonormal Basis ||  ||
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|-
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| 3.7 || *Inner Product of Functions ||  ||
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|-
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| Project 1 ||  ||  ||
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|-
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| rowspan="3" | 10 || 5.1 || Vector Calculus Intro and Taylor Series ||  ||
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|-
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| 5.1, 5.2 || Differentiation Rules Review and Partial Derivatives ||  ||
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|-
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| 5.2 || Gradients- Examples, visualizations, computation ||  ||
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|-
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| rowspan="2" | 11 || 5.3 || Gradients of Vector-Valued Functions ||  ||
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|-
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| 5.4, Dhrymes 78 || Gradients of Matrices ||  ||
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|-
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| rowspan="2" | 12 || Exam 2 ||  ||  ||  
 
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| 2 || || Vector and matrix operations: Dot and cross products, matrix transpose, determinants.  ||  ||  
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| 5.5, Dhrymes 78 || Useful Identities for Computing Gradients ||  ||  
 
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| 3 || || Vector and matrix operations: Matrix addition, multiplication and inverse. ||  ||  
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| rowspan="3" | 13 || 5.7 || Higher-Order Derivatives ||  ||  
 
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| 4 || || Cramer's rule and solutions of linear systems ||  ||  
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| Notes || Minimization via Newton's Method & Backpropagation ||  ||  
 
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| 5 ||  || Full rank, undetermined, and overdetermined systems. Least square solutions ||  ||  
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| Project 2 ||  ||  ||  
 
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| 6 || || Eigenvalues and eigenvectors. Canonical solution to linear systems of differential equations.  ||  ||  
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| rowspan="3" | 14 || 5.8 || Multivariate Taylor Series ||  ||  
 
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| 7 || || Calculus operations in vectors and matrices, i.e. how to derive a matrix with respect to a vector?  ||  ||  
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| Notes || Linear optimization: Simplex method ||  ||  
 
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| 8 ||  || Optimization: Linear problems, and nonlinear problems (constrained and unconstrained) ||  ||  
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| 7.1 || Optimization Using Gradient Descent ||  ||  
 
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| 9 || || Lagrange multiplier ||  ||  
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| rowspan="3" | 15 || 7.2 and Notes || *Constrained Optimization and Lagrange Multipliers: PCA ||  ||  
 
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| 10 || || Taylor series in one and multiple variables. Jacobians and Hessians, i.e. nabla and Laplace operators.  ||  ||  
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| Bishop, Duda et al. || Feed-forward Artificial Neural Networks ||  ||  
 
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| 11 || || Principal component analysis ||  ||  
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| Bishop, Duda et al. || Backpropagation in ANNs ||  ||  
 
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| 12 || || Gradient descent ||  ||  
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| rowspan="3" | 16 || Bishop, Duda et al. || Activation Functions: Linear & Nonlinear ||  ||  
 
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| 13 || || Neural networks as nonlinear transformations ||  ||  
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| Bishop, Duda et al. || Step-by-step simple ANN ||  ||  
 
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| 14 ||  || Implementation of a simple neural network with gradient descent
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| Final Project ||  ||  ||  
 
|}
 
|}

Latest revision as of 14:44, 24 August 2026

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

Week Section Topic Prerequisites SLOs
1 2.1 Systems of Linear Equations
2 2.2 Matrices
2.3 Solving systems of linear equations
3 2.4 Vector spaces
4 2.5 Linear Independence
2.6 Basis & Rank
5 Exam 1
2.7 Linear Mappings
6 2.7 Linear Mappings (examples)
4.1 Determinant and Traces
7 4.2 Eigenvalues & Eigenvectors
4.3, 4.4 Matrix Factorizations (Diagonalization)
8 3.1, 3.2, 3.3 Norms, Inner Products, Lengths & Distances
3.4 Angles & orthogonality
9 3.5 *Orthonormal Basis
3.7 *Inner Product of Functions
Project 1
10 5.1 Vector Calculus Intro and Taylor Series
5.1, 5.2 Differentiation Rules Review and Partial Derivatives
5.2 Gradients- Examples, visualizations, computation
11 5.3 Gradients of Vector-Valued Functions
5.4, Dhrymes 78 Gradients of Matrices
12 Exam 2
5.5, Dhrymes 78 Useful Identities for Computing Gradients
13 5.7 Higher-Order Derivatives
Notes Minimization via Newton's Method & Backpropagation
Project 2
14 5.8 Multivariate Taylor Series
Notes Linear optimization: Simplex method
7.1 Optimization Using Gradient Descent
15 7.2 and Notes *Constrained Optimization and Lagrange Multipliers: PCA
Bishop, Duda et al. Feed-forward Artificial Neural Networks
Bishop, Duda et al. Backpropagation in ANNs
16 Bishop, Duda et al. Activation Functions: Linear & Nonlinear
Bishop, Duda et al. Step-by-step simple ANN
Final Project
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