An introduction to linear algebra and Fourier analysis, emphasizing geometric intuition and connections between ideas. Materials below are organized by semester.

Course outline

Before the midterm

  • Row operations and why they preserve the solution set
  • Gaussian elimination and Gauss–Jordan elimination
  • General solutions of linear systems and column combinations
  • Vector spaces, subspaces, linear combinations, independence, span, basis, and dimension
  • Linear transformations and the four fundamental subspaces

After the midterm

  • Eigenvalues, eigenvectors, and diagonalization
  • Inner products and the Gram–Schmidt process
  • Periodic functions and Fourier series, including half-range sine and cosine series
  • Complex Fourier series, Fourier transforms, and sine and cosine transforms

Download the complete course outline.

Lecture notes

The formula trees used in class are in chapters 11 and 12 of the lecture notes.

Further explorations

$$ \begin{aligned} \mathbf{x}'(t)&=M\mathbf{x}(t),\\ \mathbf{x}(t)&=\exp(tM)\mathbf{x}(0),\\ M&=\begin{bmatrix}a&b\\c&d\end{bmatrix}. \end{aligned} $$

Summer 2024 materials

AssessmentMaterial
Before the midtermAssignment 1 · Quiz 1 · Quiz 1 solutions
Midterm preparationPractice problems
After the midtermAssignment 2
Final preparationPractice problems

Summer 2024 Discord

Spring 2025 materials

AssessmentMaterial
Before the midtermAssignment 1 · Quiz 1 with solutions · Assignment 2
Midterm preparationPractice problems
After the midtermAssignment 3 · Quiz 2 with solutions
Final preparationPractice problems

Spring 2025 Discord