Difference between revisions of "MAT2253"

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(Create MAT2253 to remove 2243 (number unavailable).)
 
(→‎List of Topics: Edited by Vu Hoang, Stephen Peña, Juan B. Gutiérrez)
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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.
 
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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==List of Topics==
 
  
 
{| class="wikitable"
 
{| class="wikitable"
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! Week !! Section !! Topic !! Prerequisites !! SLOs
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! Session !! 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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| 2 || || Vector and matrix operations: Dot and cross products, matrix transpose, determinants.  ||  ||  
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| 2 || 2.2 || Matrices ||  ||  
 
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| 3 || || Vector and matrix operations: Matrix addition, multiplication and inverse. ||  ||  
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| 3 || 2.3 || Solving systems of linear equations ||  ||  
 
|-
 
|-
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| 4 || || Cramer's rule and solutions of linear systems ||  ||  
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| 4 || 3.1, 3.2, 3.3 || Norms, Inner Products, Lengths & Distances ||  ||  
 
|-
 
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| 5 || || Full rank, undetermined, and overdetermined systems. Least square solutions ||  ||  
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| 5 || 3.4 || Angles & orthogonality ||  ||  
 
|-
 
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| 6 || || Eigenvalues and eigenvectors. Canonical solution to linear systems of differential equations.  ||  ||  
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| 6 || 2.4, 2.5 || Vector spaces & Linear Independendence ||  ||  
 
|-
 
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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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| 7 || Mini-test || ||  ||  
 
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| 8 || || Optimization: Linear problems, and nonlinear problems (constrained and unconstrained) ||  ||  
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| 8 || 2.6 || Basis & Rank ||  ||  
 
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| 9 || || Lagrange multiplier ||  ||  
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| 9 || 2.7 || Linear Mappings ||  ||  
 
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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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| 10 || 4.1 || Determinant and Traces ||  ||  
 
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| 11 || || Principal component analysis ||  ||  
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| 11 || 4.2 || Eigenvalues & Eigenvectors ||  ||  
 
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| 12 || || Gradient descent ||  ||  
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| 12 || 4.3, 4.4 || Matrix Factorization ||  ||  
 
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| 13 || || Neural networks as nonlinear transformations ||  ||  
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| 13 || 3.5 || Orthonormal Basis ||  ||  
 
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| 14 ||  || Implementation of a simple neural network with gradient descent
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| 14 || 3.7 || Inner Product of Functions ||  ||
 +
|-
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| 15 || 3.9 || Rotations ||  ||
 +
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| 16 || Mini-test ||  ||  ||
 +
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| 17 || 5.1 || Vector Calculus ||  ||
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| 18 || 5.1 || Taylor Series ||  ||  
 +
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| 19 || 5.1 || Differentiation Rules Review ||  ||
 +
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| 20 || 5.2 || Partial Derivatives ||  ||
 +
|-
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| 21 || 5.2 || Gradients - Examples, visualizations, computaiton ||  ||
 +
|-
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| 22 || 5.2 || Rules for Partial Differentiation & Chain Rule ||  ||
 +
|-
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| 23 || Mini-test ||  ||  ||
 +
|-
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| 24 || 5.3 || Gradients of Vector-Valued Functions ||  ||
 +
|-
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| 25 || 5.3 || Gradients of Vector-Valued Functions ||  ||
 +
|-
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| 26 || 5.4, Dhrymes 78 || Gradients of Matrices ||  ||
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| 27 || 5.5, Dhrymes 78 || Useful Identities for Computing Gradients ||  ||
 +
|-
 +
| 28 || 5.7 || Higher-Order Derivatives ||  ||
 +
|-
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| 29 || Notes || Minimization via Newton's Method & Backpropagation ||  ||
 +
|-
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| 30 || Min-test ||  ||  ||
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|-
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| 31 || 5.8 || Linearization & Multivariate Taylor Series ||  ||
 +
|-
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| 32 || Notes || Linear optimization: Simplex method ||  ||
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|-
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| 33 || 7.1 || Optimization Using Gradient Descent ||  ||
 +
|-
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| 34 || 7.2 || Constrained Optimization and Lagrange Multipliers ||  ||
 +
|-
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| 35 || 7.3 || Convex Optimization ||  ||
 +
|-
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| 36 || Mini-Test ||  ||  ||
 +
|-
 +
| 37 || Bishop, Duda et al. || Feed-forward Artificial Neural Networks ||  ||
 +
|-
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| 38 || Bishop, Duda et al. || Backpropagation in ANNs ||  ||
 +
|-
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| 39 || Bishop, Duda et al. || Activation Functions: Linear & Nonlinear ||  ||
 +
|-
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| 40 || Bishop, Duda et al. || Step-by-step simple ANN ||  ||
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| 41 || Bishop, Duda et al. || Measures of performance ||  ||
 +
|-
 +
| 42 || Bishop, Duda et al. || More complex architectures of ANNs ||  ||
 +
|-
 +
| 43 || Mini-test ||  ||  ||
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|-
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| 44 || Final project ||  ||  ||
 +
|-
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| 45 || Review ||  ||  ||
 
|}
 
|}

Revision as of 12:59, 28 January 2025

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.

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
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