MAT2253 - Revision history

Stephen.pena: moved matrix factorization up

2026-03-17T16:09:06Z

moved matrix factorization up

Stephen.pena: switched the order of determinants and inner product coverage

2026-03-17T13:16:20Z

switched the order of determinants and inner product coverage

Stephen.pena at 21:09, 28 October 2025

2025-10-28T21:09:13Z

Stephen.pena at 15:55, 19 September 2025

2025-09-19T15:55:19Z

Juan.gutierrez3: /* Applied Linear Algebra */ Added textbooks

2025-01-29T04:02:32Z

Applied Linear Algebra: Added textbooks

Juan.gutierrez3: /* List of Topics */ Edited by Vu Hoang, Stephen Peña, Juan B. Gutiérrez

2025-01-28T18:59:10Z

List of Topics: Edited by Vu Hoang, Stephen Peña, Juan B. Gutiérrez

Eduardo.duenez: Create MAT2253 to remove 2243 (number unavailable).

2023-04-01T23:31:01Z

Create MAT2253 to remove 2243 (number unavailable).

New page

==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 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.

==List of Topics==

{| class="wikitable"
! Week !! Section !! Topic !! Prerequisites !! SLOs
|-
| 1 || || Notions of linear systems of equations to introduce the concepts of vector and matrices. || ||
|-
| 2 || || Vector and matrix operations: Dot and cross products, matrix transpose, determinants. || ||
|-
| 3 || || Vector and matrix operations: Matrix addition, multiplication and inverse. || ||
|-
| 4 || || Cramer's rule and solutions of linear systems || ||
|-
| 5 || || Full rank, undetermined, and overdetermined systems. Least square solutions || ||
|-
| 6 || || Eigenvalues and eigenvectors. Canonical solution to linear systems of differential equations. || ||
|-
| 7 || || Calculus operations in vectors and matrices, i.e. how to derive a matrix with respect to a vector? || ||
|-
| 8 || || Optimization: Linear problems, and nonlinear problems (constrained and unconstrained) || ||
|-
| 9 || || Lagrange multiplier || ||
|-
| 10 || || Taylor series in one and multiple variables. Jacobians and Hessians, i.e. nabla and Laplace operators. || ||
|-
| 11 || || Principal component analysis || ||
|-
| 12 || || Gradient descent || ||
|-
| 13 || || Neural networks as nonlinear transformations || ||
|-
| 14 || || Implementation of a simple neural network with gradient descent
|}

@@ Line 34: / Line 34: @@
 | 11 || 4.2 || Eigenvalues & Eigenvectors ||  ||
 |-
-| 12 || 3.1, 3.2, 3.3 || Norms, Inner Products, Lengths & Distances ||  ||
+| 12 || 4.3, 4.4 || Matrix Factorization ||  ||
 |-
-| 13 || 3.4 || Angles & orthogonality ||  ||
+| 13 || 3.1, 3.2, 3.3 || Norms, Inner Products, Lengths & Distances ||  ||
 |-
-| 14 || 3.5 || Orthonormal Basis ||  ||
+| 14 || 3.4 || Angles & orthogonality ||  ||
 |-
-| 15 || 3.7 || Inner Product of Functions ||  ||
+| 15 || 3.5 || Orthonormal Basis ||  ||
 |-
-| 16 || Project 1 ||  ||  ||
+| 16 || 3.7 || Inner Product of Functions ||  ||
 |-
-| 17 || 4.3, 4.4 || Matrix Factorization ||  ||
+| 17 || Project 1 ||  ||  ||
 |-
 | 18 || 5.1 ||Vector Calculus Intro and Taylor Series ||  ||

@@ Line 30: / Line 30: @@
 | 9 || 2.7 || Linear Mappings (examples) ||  ||
 |-
-| 10 || 3.1, 3.2, 3.3 || Norms, Inner Products, Lengths & Distances ||  ||
+| 10 || 4.1 || Determinant and Traces ||  ||
 |-
-| 11 || 3.4 || Angles & orthogonality ||  ||
+| 11 || 4.2 || Eigenvalues & Eigenvectors ||  ||
 |-
-| 12 || 3.5 || Orthonormal Basis ||  ||
+| 12 || 3.1, 3.2, 3.3 || Norms, Inner Products, Lengths & Distances ||  ||
 |-
-| 13 || 3.7 || Inner Product of Functions ||  ||
+| 13 || 3.4 || Angles & orthogonality ||  ||
 |-
-| 14 || Project 1 ||  ||  ||
+| 14 || 3.5 || Orthonormal Basis ||  ||
 |-
-| 15 || 4.1 || Determinant and Traces ||  ||
+| 15 || 3.7 || Inner Product of Functions ||  ||
 |-
-| 16 || 4.2 || Eigenvalues & Eigenvectors ||  ||
+| 16 || Project 1 ||  ||  ||
 |-
 | 17 || 4.3, 4.4 || Matrix Factorization ||  ||

@@ Line 60: / Line 60: @@
 | 24 || 5.5, Dhrymes 78 || Useful Identities for Computing Gradients ||  ||
 |-
-| 25 || 5.3 || Gradients of Vector-Valued Functions ||  ||
+| 25 || 5.7 || Higher-Order Derivatives ||  ||
 |-
-| 26 || 5.4, Dhrymes 78 || Gradients of Matrices ||  ||
+| 26 || Notes || Minimization via Newton's Method & Backpropagation ||  ||
 |-
-| 27 || 5.5, Dhrymes 78 || Useful Identities for Computing Gradients ||  ||
+| 27 ||Project 2 ||  ||  ||
 |-
-| 28 || 5.7 || Higher-Order Derivatives ||  ||
+| 28 || 5.8 ||  Multivariate Taylor Series ||  ||
 |-
-| 29 || Notes || Minimization via Newton's Method & Backpropagation ||  ||
+| 29 || Notes || Linear optimization: Simplex method ||  ||
 |-
-| 30 ||Project 2 ||  ||  ||
+| 30 || 7.1 || Optimization Using Gradient Descent ||  ||
 |-
-| 31 || 5.8 ||  Multivariate Taylor Series ||  ||
+| 31|| 7.2 and Notes || Constrained Optimization and Lagrange Multipliers: PCA ||  ||
 |-
-| 32 || Notes || Linear optimization: Simplex method ||  ||
+| 32 || 7.3 || Convex Optimization (time permitting) ||  ||
 |-
-| 33 || 7.1 || Optimization Using Gradient Descent ||  ||
+| 33 || Bishop, Duda et al. || Feed-forward Artificial Neural Networks ||  ||
 |-
-| 34 || 7.2 and Notes || Constrained Optimization and Lagrange Multipliers: PCA ||  ||
+| 34 || Bishop, Duda et al. || Backpropagation in ANNs ||  ||
 |-
-| 35 || 7.3 || Convex Optimization (time permitting) ||  ||
+| 35 || Bishop, Duda et al. || Activation Functions: Linear & Nonlinear ||  ||
 |-
-| 36 || Bishop, Duda et al. || Feed-forward Artificial Neural Networks ||  ||
+| 36 || Bishop, Duda et al. || Step-by-step simple ANN ||  ||
 |-
-| 37 || Bishop, Duda et al. || Backpropagation in ANNs ||  ||
+| 37 || Bishop, Duda et al. || Measures of performance ||  ||
 |-
-| 38 || Bishop, Duda et al. || Activation Functions: Linear & Nonlinear ||  ||
+| 38 || Bishop, Duda et al. || More complex architectures of ANNs ||  ||
 |-
-| 39 || Bishop, Duda et al. || Step-by-step simple ANN ||  ||
+| 39 || Final Project Introduction ||  ||  ||
 |-
-| 40 || Bishop, Duda et al. || Measures of performance ||  ||
+| 40 || Final project||  ||  ||
 |-
-| 41 || Bishop, Duda et al. || More complex architectures of ANNs ||  ||
+| 41 || Review ||  ||  ||
 |}

@@ Line 18: / Line 18: @@
 | 3 || 2.3 || Solving systems of linear equations ||  ||
 |-
-| 4 || 3.1, 3.2, 3.3 || Norms, Inner Products, Lengths & Distances ||  ||
+| 4 || 2.4 || Vector spaces ||  ||
 |-
-| 5 || 3.4 || Angles & orthogonality ||  ||
+| 5 || 2.5 ||Linear Independence ||  ||
 |-
-| 6 || 2.4, 2.5 || Vector spaces & Linear Independendence ||  ||
+| 6 || 2.6 || Basis & Rank ||  ||
 |-
-| 7 || Mini-test ||  ||  ||
+| 7 || Exam 1 ||  ||  ||
 |-
-| 8 || 2.6 || Basis & Rank ||  ||
+| 8 || 2.7 || Linear Mappings ||  ||
 |-
-| 9 || 2.7 || Linear Mappings ||  ||
+| 9 || 2.7 || Linear Mappings (examples) ||  ||
 |-
-| 10 || 4.1 || Determinant and Traces ||  ||
+| 10 || 3.1, 3.2, 3.3 || Norms, Inner Products, Lengths & Distances ||  ||
 |-
-| 11 || 4.2 || Eigenvalues & Eigenvectors ||  ||
+| 11 || 3.4 || Angles & orthogonality ||  ||
 |-
-| 12 || 4.3, 4.4 || Matrix Factorization ||  ||
+| 12 || 3.5 || Orthonormal Basis ||  ||
 |-
-| 13 || 3.5 || Orthonormal Basis ||  ||
+| 13 || 3.7 || Inner Product of Functions ||  ||
 |-
-| 14 || 3.7 || Inner Product of Functions ||  ||
+| 14 || Project 1 ||  ||  ||
 |-
-| 15 || 3.9 || Rotations ||  ||
+| 15 || 4.1 || Determinant and Traces ||  ||
 |-
-| 16 || Mini-test ||  ||  ||
+| 16 || 4.2 || Eigenvalues & Eigenvectors ||  ||
 |-
-| 17 || 5.1 || Vector Calculus ||  ||
+| 17 || 4.3, 4.4 || Matrix Factorization ||  ||
 |-
-| 18 || 5.1 || Taylor Series ||  ||
+| 18 || 5.1 ||Vector Calculus Intro and Taylor Series ||  ||
 |-
-| 19 || 5.1 || Differentiation Rules Review ||  ||
+| 19 || 5.1, 5.2 || Differentiation Rules Review and Partial Derivatives ||  ||
 |-
-| 20 || 5.2 || Partial Derivatives ||  ||
+| 20 || 5.2 || Gradients- Examples, visualizations, computation ||  ||
 |-
-| 21 || 5.2 || Gradients - Examples, visualizations, computaiton ||  ||
+| 21 || 5.3 || Gradients of Vector-Valued Functions ||  ||
 |-
-| 22 || 5.2 || Rules for Partial Differentiation & Chain Rule ||  ||
+| 22 || 5.4, Dhrymes 78 || Gradients of Matrices ||  ||
 |-
-| 23 || Mini-test ||  ||  ||
+| 23 || Exam 2 ||  ||  ||
 |-
-| 24 || 5.3 || Gradients of Vector-Valued Functions ||  ||
+| 24 || 5.5, Dhrymes 78 || Useful Identities for Computing Gradients ||  ||
 |-
 | 25 || 5.3 || Gradients of Vector-Valued Functions ||  ||
@@ Line 70: / Line 70: @@
 | 29 || Notes || Minimization via Newton's Method & Backpropagation ||  ||
 |-
-| 30 || Min-test ||  ||  ||
+| 30 ||Project 2 ||  ||  ||
 |-
-| 31 || 5.8 || Linearization & Multivariate Taylor Series ||  ||
+| 31 || 5.8 ||  Multivariate Taylor Series ||  ||
 |-
 | 32 || Notes || Linear optimization: Simplex method ||  ||
@@ Line 78: / Line 78: @@
 | 33 || 7.1 || Optimization Using Gradient Descent ||  ||
 |-
-| 34 || 7.2 || Constrained Optimization and Lagrange Multipliers ||  ||
+| 34 || 7.2 and Notes || Constrained Optimization and Lagrange Multipliers: PCA ||  ||
 |-
-| 35 || 7.3 || Convex Optimization ||  ||
+| 35 || 7.3 || Convex Optimization (time permitting) ||  ||
 |-
-| 36 || Mini-Test ||  ||  ||
+| 36 || Bishop, Duda et al. || Feed-forward Artificial Neural Networks ||  ||
 |-
-| 37 || Bishop, Duda et al. || Feed-forward Artificial Neural Networks ||  ||
+| 37 || Bishop, Duda et al. || Backpropagation in ANNs ||  ||
 |-
-| 38 || Bishop, Duda et al. || Backpropagation in ANNs ||  ||
+| 38 || Bishop, Duda et al. || Activation Functions: Linear & Nonlinear ||  ||
 |-
-| 39 || Bishop, Duda et al. || Activation Functions: Linear & Nonlinear ||  ||
+| 39 || Bishop, Duda et al. || Step-by-step simple ANN ||  ||
 |-
-| 40 || Bishop, Duda et al. || Step-by-step simple ANN ||  ||
+| 40 || Bishop, Duda et al. || Measures of performance ||  ||
 |-
-| 41 || Bishop, Duda et al. || Measures of performance ||  ||
+| 41 || Bishop, Duda et al. || More complex architectures of ANNs ||  ||
 |-
-| 42 || Bishop, Duda et al. || More complex architectures of ANNs ||  ||
+| 42 || Final Project Introduction ||  ||  ||
 |-
-| 43 || Mini-test ||  ||  ||
+| 43 || Final project||  ||  ||
 |-
-| 44 || Final project ||  ||  ||
+| 44 || Review ||  ||  ||
 |}

@@ Line 3: / Line 3: @@
 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.
 {| class="wikitable"

@@ Line 4: / Line 4: @@
 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.
 {| class="wikitable"
-! Week !! Section !! Topic !! Prerequisites !! SLOs
+! Session !! Section !! Topic !! Prerequisites !! SLOs
 |-
-| 1 ||  || Notions of linear systems of equations to introduce the concepts of vector and matrices.   ||  ||
+| 1 || 2.1 || Systems of Linear Equations ||  ||
 |-
-| 2 ||  || Vector and matrix operations: Dot and cross products, matrix transpose, determinants.  ||  ||
+| 2 || 2.2 || Matrices ||  ||
 |-
-| 3 ||  || Vector and matrix operations: Matrix addition, multiplication and inverse. ||  ||
+| 3 || 2.3 || Solving systems of linear equations ||  ||
 |-
-| 4 ||  || Cramer's rule and solutions of linear systems ||  ||
+| 4 || 3.1, 3.2, 3.3 || Norms, Inner Products, Lengths & Distances ||  ||
 |-
-| 5 ||  || Full rank, undetermined, and overdetermined systems. Least square solutions ||  ||
+| 5 || 3.4 || Angles & orthogonality ||  ||
 |-
-| 6 ||  || Eigenvalues and eigenvectors. Canonical solution to linear systems of differential equations.  ||  ||
+| 6 || 2.4, 2.5 || Vector spaces & Linear Independendence ||  ||
 |-
-| 7 ||  || Calculus operations in vectors and matrices, i.e. how to derive a matrix with respect to a vector?  ||  ||
+| 7 || Mini-test ||  ||  ||
 |-
-| 8 ||  || Optimization: Linear problems, and nonlinear problems (constrained and unconstrained) ||  ||
+| 8 || 2.6 || Basis & Rank ||  ||
 |-
-| 9 ||  || Lagrange multiplier ||  ||
+| 9 || 2.7 || Linear Mappings ||  ||
 |-
-| 10 ||  || Taylor series in one and multiple variables. Jacobians and Hessians, i.e. nabla and Laplace operators.  ||  ||
+| 10 || 4.1 || Determinant and Traces ||  ||
 |-
-| 11 ||  || Principal component analysis ||  ||
+| 11 || 4.2 || Eigenvalues & Eigenvectors ||  ||
 |-
-| 12 ||  || Gradient descent ||  ||
+| 12 || 4.3, 4.4 || Matrix Factorization ||  ||
 |-
-| 13 ||  || Neural networks as nonlinear transformations ||  ||
+| 13 || 3.5 || Orthonormal Basis ||  ||
 |-
-| 14 ||  || Implementation of a simple neural network with gradient descent
+| 14 || 3.7 || Inner Product of Functions ||  ||
 |}