Difference between revisions of "MAT2253"

From Department of Mathematics at UTSA
Jump to navigation Jump to search
(→‎List of Topics: Edited by Vu Hoang, Stephen Peña, Juan B. Gutiérrez)
m (marked optional topics Fall 2026)
 
(6 intermediate revisions by 2 users not shown)
Line 3: Line 3:
 
Prerequisite: [[MAT1214]]/[[MAT1213]] Calculus I
 
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 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.
+
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
  
 
{| class="wikitable"
 
{| class="wikitable"
−
! Session !! Section !! Topic !! Prerequisites !! SLOs
+
! Week !! Section !! Topic !! Prerequisites !! SLOs
 
|-
 
|-
 
| 1 || 2.1 || Systems of Linear Equations ||  ||  
 
| 1 || 2.1 || Systems of Linear Equations ||  ||  
 
|-
 
|-
−
| 2 || 2.2 || Matrices ||  ||
+
| rowspan="2" | 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 ||  ||  
+
| 2.3 || Solving systems of linear equations ||  ||  
 
|-
 
|-
−
| 13 || 3.5 || Orthonormal Basis ||  ||  
+
| 3 || 2.4 || Vector spaces ||  ||  
 
|-
 
|-
−
| 14 || 3.7 || Inner Product of Functions ||  ||  
+
| rowspan="2" | 4 || 2.5 || Linear Independence ||  ||  
 
|-
 
|-
−
| 15 || 3.9 || Rotations ||  ||  
+
| 2.6 || Basis & Rank ||  ||  
 
|-
 
|-
−
| 16 || Mini-test ||  ||  ||  
+
| rowspan="2" | 5 || Exam 1 ||  ||  ||  
 
|-
 
|-
−
| 17 || 5.1 || Vector Calculus ||  ||  
+
| 2.7 || Linear Mappings ||  ||  
 
|-
 
|-
−
| 18 || 5.1 || Taylor Series ||  ||  
+
| rowspan="2" | 6 || 2.7 || Linear Mappings (examples) ||  ||  
 
|-
 
|-
−
| 19 || 5.1 || Differentiation Rules Review ||  ||  
+
| 4.1 || Determinant and Traces ||  ||  
 
|-
 
|-
−
| 20 || 5.2 || Partial Derivatives ||  ||  
+
| rowspan="2" | 7 || 4.2 || Eigenvalues & Eigenvectors ||  ||  
 
|-
 
|-
−
| 21 || 5.2 || Gradients - Examples, visualizations, computaiton ||  ||  
+
| 4.3, 4.4 || Matrix Factorizations (Diagonalization) ||  ||  
 
|-
 
|-
−
| 22 || 5.2 || Rules for Partial Differentiation & Chain Rule ||  ||  
+
| rowspan="2" | 8 || 3.1, 3.2, 3.3 || Norms, Inner Products, Lengths & Distances ||  ||  
 
|-
 
|-
−
| 23 || Mini-test ||  ||  ||  
+
| 3.4 || Angles & orthogonality ||  ||  
 
|-
 
|-
−
| 24 || 5.3 || Gradients of Vector-Valued Functions ||  ||  
+
| rowspan="3" | 9 || 3.5 || *Orthonormal Basis ||  ||  
 
|-
 
|-
−
| 25 || 5.3 || Gradients of Vector-Valued Functions ||  ||  
+
| 3.7 || *Inner Product of Functions ||  ||  
 
|-
 
|-
−
| 26 || 5.4, Dhrymes 78 || Gradients of Matrices ||  ||  
+
| Project 1 || ||  ||  
 
|-
 
|-
−
| 27 || 5.5, Dhrymes 78 || Useful Identities for Computing Gradients ||  ||  
+
| rowspan="3" | 10 || 5.1 || Vector Calculus Intro and Taylor Series ||  ||  
 
|-
 
|-
−
| 28 || 5.7 || Higher-Order Derivatives ||  ||  
+
| 5.1, 5.2 || Differentiation Rules Review and Partial Derivatives ||  ||  
 
|-
 
|-
−
| 29 || Notes || Minimization via Newton's Method & Backpropagation ||  ||  
+
| 5.2 || Gradients- Examples, visualizations, computation ||  ||  
 
|-
 
|-
−
| 30 || Min-test || ||  ||  
+
| rowspan="2" | 11 || 5.3 || Gradients of Vector-Valued Functions ||  ||  
 
|-
 
|-
−
| 31 || 5.8 || Linearization & Multivariate Taylor Series ||  ||  
+
| 5.4, Dhrymes 78 || Gradients of Matrices ||  ||  
 
|-
 
|-
−
| 32 || Notes || Linear optimization: Simplex method ||  ||  
+
| rowspan="2" | 12 || Exam 2 ||  ||  ||  
 
|-
 
|-
−
| 33 || 7.1 || Optimization Using Gradient Descent ||  ||  
+
| 5.5, Dhrymes 78 || Useful Identities for Computing Gradients ||  ||  
 
|-
 
|-
−
| 34 || 7.2 || Constrained Optimization and Lagrange Multipliers ||  ||  
+
| rowspan="3" | 13 || 5.7 || Higher-Order Derivatives ||  ||  
 
|-
 
|-
−
| 35 || 7.3 || Convex Optimization ||  ||  
+
| Notes || Minimization via Newton's Method & Backpropagation ||  ||  
 
|-
 
|-
−
| 36 || Mini-Test ||  ||  ||  
+
| Project 2 ||  ||  ||  
 
|-
 
|-
−
| 37 || Bishop, Duda et al. || Feed-forward Artificial Neural Networks ||  ||  
+
| rowspan="3" | 14 || 5.8 || Multivariate Taylor Series ||  ||  
 
|-
 
|-
−
| 38 || Bishop, Duda et al. || Backpropagation in ANNs ||  ||  
+
| Notes || Linear optimization: Simplex method ||  ||  
 
|-
 
|-
−
| 39 || Bishop, Duda et al. || Activation Functions: Linear & Nonlinear ||  ||  
+
| 7.1 || Optimization Using Gradient Descent ||  ||  
 
|-
 
|-
−
| 40 || Bishop, Duda et al. || Step-by-step simple ANN ||  ||  
+
| rowspan="3" | 15 || 7.2 and Notes || *Constrained Optimization and Lagrange Multipliers: PCA ||  ||  
 
|-
 
|-
−
| 41 || Bishop, Duda et al. || Measures of performance ||  ||  
+
| Bishop, Duda et al. || Feed-forward Artificial Neural Networks ||  ||  
 
|-
 
|-
−
| 42 || Bishop, Duda et al. || More complex architectures of ANNs ||  ||  
+
| Bishop, Duda et al. || Backpropagation in ANNs ||  ||  
 
|-
 
|-
−
| 43 || Mini-test || ||  ||  
+
| rowspan="3" | 16 || Bishop, Duda et al. || Activation Functions: Linear & Nonlinear ||  ||  
 
|-
 
|-
−
| 44 || Final project ||  ||  ||  
+
| Bishop, Duda et al. || Step-by-step simple ANN ||  ||  
 
|-
 
|-
−
| 45 || Review ||  ||  ||  
+
| 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
Retrieved from "https://mathresearch.utsa.edu/wiki/index.php?title=MAT2253&oldid=5456"