Linear Algebra — Teaching Materials

This repository contains three components for the Spring 2025 Linear Algebra course that I devised and piloted as a Teaching Assistant:

1. Exercise 1: Linear Transformations & Coordinate Systems
2. Exercise 2: Orthogonal Diagonalization, Quadratic Forms, and Constrained Optimization
3. Project: Latent-Factor–Based Recommender

Each component has its own detailed README and notebook(s). The exercises provide geometric intuition and foundational tools; the project applies these tools to build and evaluate a recommender system grounded in linear algebra.

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Components

1) Exercise 1 — Linear Transformations, Coordinate Systems, and Rank Geometry

• Notebooks:
  • ex1-linear-transformations-and-coordinate-systems.ipynb (instructor/piloted)
  • ex1-linear-transformations-and-coordinate-systems_raw.ipynb (student/raw)
• Focus:
  • Matrices as linear maps; geometric effects (rotation, shear, stretch, projection)
  • Vectors in different coordinate systems; change of basis via a basis matrix
  • Rank and the geometry of images (collapse to point/line/plane, dimension preservation)

2) Exercise 2 — Orthogonal Diagonalization, Quadratic Forms, and Constrained Optimization

• Notebooks:
  • ex2-orthogonal-diagonalization-quadratic-forms-optimization.ipynb (instructor/piloted)
  • ex2-orthogonal-diagonalization-quadratic-forms-optimization_raw.ipynb (student/raw)
• Focus:
  • Orthogonal diagonalization of symmetric matrices and reconstruction error (PCA intuition)
  • Quadratic forms and contour analysis in eigen vs. Cartesian coordinates
  • Constrained optimization (Rayleigh quotient, Lagrange multipliers; multiple constraints)

3) Project — Latent-Factor–Based Recommender

• Notebook:
  • project-latent-factor-based-recommender.ipynb (instructor/piloted)
• Focus:
  • User–item matrix modeling; cosine similarity (user–user / item–item)
  • Matrix factorization with regularization and bias terms; rating prediction & top-N ranking
  • Latent factor based analysis and basic clustering in latent space.

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Intended Audience and Usage

• Students: start with the _raw versions and follow prompts; compare to the piloted versions after completion.
• Instructors/TAs: use piloted versions for reference solutions, additional diagnostics, and visualizations.