Advanced Linear Algebra Course

Course Description

Welcome to this advanced Linear Algebra course, where you'll dive deep into the fascinating world of determinants, eigenvalues, and eigenvectors. This course is designed to expand your understanding of linear transformations and their applications in various fields, including computer graphics, discrete dynamical systems, and Markov chains. You'll explore how these concepts are utilized in industry, science, engineering, and mathematics, providing you with valuable skills that are highly sought after in today's job market.

What Students Will Learn

• Computing determinants using various methods
• Calculating areas of regions under linear transformations
• Modeling and solving real-world problems with Markov chains
• Identifying and verifying eigenvectors and eigenvalues
• Constructing eigenspaces for matrices
• Characterizing matrix invertibility using determinants and eigenvalues
• Factorizing matrices with complex eigenvalues
• Analyzing linear transformations in terms of rotation and dilation
• Applying matrix powers to understand long-term behavior of Markov chains
• Constructing and analyzing transition matrices, Markov chains, and Google matrices for web page ranking

Prerequisites

To get the most out of this course, it's recommended that you have completed the following:

• Linear Algebra II course (part of the four-part linear algebra sequence on edX)
• A solid understanding of linear equations and matrix algebra

Course Content

• Determinants and their applications
• Invertibility criteria for square matrices
• Area measurement under linear transformations
• Eigenvalues and eigenvectors
• Discrete dynamical systems
• Markov chains
• Complex eigenvalues and matrix factorization
• PageRank algorithm and Google matrices

Who This Course Is For

• Students pursuing advanced mathematics or engineering degrees
• Professionals in fields such as data science, computer graphics, or machine learning
• Anyone looking to deepen their understanding of linear algebra and its real-world applications

Real-World Applications

• Computer Graphics: Understanding how linear transformations affect areas is crucial for creating realistic 3D renderings and animations.
• Data Science: Eigenvalues and eigenvectors are fundamental in dimensionality reduction techniques like Principal Component Analysis (PCA).
• Machine Learning: Many algorithms rely on concepts from linear algebra, including neural networks and support vector machines.
• Economics and Finance: Markov chains are used to model and predict market behaviors and economic trends.
• Web Search Engines: The PageRank algorithm, which you'll learn about, is the foundation of Google's search engine.
• Quantum Mechanics: Eigenvalue problems are essential in understanding quantum systems.

Course Syllabus

1. Introduction to determinants
2. Applications of determinants in area measurement and invertibility
3. Eigenvalues and eigenvectors: concepts and calculations
4. Applications of eigenvalues and eigenvectors in discrete dynamical systems
5. Markov chains: modeling and analysis
6. Complex eigenvalues and matrix factorization
7. Advanced applications: PageRank algorithm and Google matrices