Linear Algebra and Its Applications

Course Description

We cover systems of linear equations, subspaces and coordinates, linear transformations, determinants, eigenvalues and eigenvectors, diagonalization, symmetric matrices and the singular value decomposition, and orthogonal projection. The last two lectures introduce multivariable functions and the gradient.

We use Python projects to understand the mathematics behind several applications, including extracting pitch information from audio files with the Discrete Fourier Transform, image compression with the singular value decomposition, reconstructing images with eigenfaces, and the math behind the neural networks that make AI possible.

This page is a work in progress. We add notes and files as the course is taught.

Lecture Notes

1.Systems of Linear Equations

2.Geometry of Solutions

3.Solutions

4.Subspaces

5.Coordinates

6.Transformations

7.Linear Transformations

8.Composition of Transformations

9.Determinants

10.Eigenvalues and Eigenvectors

11.Similarity and Diagonalization

12.Symmetric Matrices and SVD

13.Orthogonal Projection

14.Multivariable Functions

15.Directional Derivatives and the Gradient

Projects

The Discrete Fourier Transform

The mathematics behind JPEG, MP3 and voice recognition, applied to audio files in Python. The full course resources page can be found here.

Eigenfaces

Image Compression with the SVD

Image compression using the singular value decomposition.