Lesson 15: Applications & Course Wrap-Up

1. Real-World Applications of Linear Algebra

Everything we learned is used everywhere:

• Computer Graphics: rotation, scaling, projection matrices
• Machine Learning: data as vectors, PCA (eigenvectors), linear regression (least squares)
• Physics: forces, velocities, torque (cross product), quantum mechanics (eigenstates)
• Engineering: circuits, structures, control systems (eigenvalues for stability)
• Data Science: dimensionality reduction, recommendation systems

2. Eigenvalues in Real Life

Eigenvalues tell us the "natural frequencies" or "growth rates" of systems.

Examples:

• Vibration analysis: natural frequencies of bridges
• Google PageRank: dominant eigenvector gives importance ranking
• Population models: long-term growth rates
• PCA: directions of maximum variance in data

3. Course Wrap-Up & Next Steps

You now know the core of linear algebra:

• Vectors: representation, operations, magnitude, dot/cross product
• Matrices: multiplication, transformations
• Systems of equations: Gaussian elimination, rank, inverses
• Eigenvalues/eigenvectors: special directions & scaling

Next steps:

• Practice more systems & 3×3 matrices
• Learn SVD (Singular Value Decomposition)
• Explore applications: PCA, PageRank, computer vision
• Recommended books: "Linear Algebra Done Right" (Axler), "Introduction to Linear Algebra" (Strang)

Congratulations — you’ve built a strong foundation!