Linear Algebra

Author

Alvaro Arias

Introduction to Linear Algebra

These are the companion notes to the Winter 2025 Linear Algebra class. The course is designed to bridge theoretical understanding with practical applications, recognizing two fundamental aspects of linear algebra:

  1. The geometric/visual interpretation of linear algebra concepts, helping students move beyond purely algorithmic approaches to understand systems as relationships between vectors and transformations that can be visualized.
  2. The integration of computational tools to handle routine calculations, allowing focus on interpretation and real-world applications rather than manual computation.

Linear algebra serves as the mathematical foundation for numerous real-world applications that shape our modern world. You’ll encounter it in:

  • Sciences & Engineering: Including quantum computing, computational biology, signal processing, robotics and autonomous systems
  • Computing, Data & Statistics: Including machine learning, computer graphics, statistical analysis, and network modeling
  • Business & Economics: Including market modeling, optimization, and equilibrium analysis

You’ll learn to view linear algebra through three lenses: theoretical foundations, geometric intuition, and practical applications - preparing you to apply these concepts across mathematics, sciences, engineering, and beyond.