Level 7.3 – Vectors and matrices

These notes are based on lectures by Angela Capel Cuevas in the academic 2025-2026. Mistakes are almost surely mine and this is not official nor a recreation of what was in the lectures.

Summary of the course [Expand]

• Lectures 1-2 are about some complex number properties - mostly a review. My notes include a topological proof of the fundamental theorem of algebra in this section.
• Lecture 3 is about the definition of vector spaces and norms.
• Lectures 4-6 are about the cauchy-schwartz inequality and a few definitions and properties of dot (scalar) and cross (Vector) products.
• Lectures 6-7 are about the delta and epsilon symbols and the summation convention.
• Lecture 8 is about linear independence and dimension - my notes prove this is well defined.
• Lectures 9-15 redefine matrices as linear maps, determinant in terms of the levi civita epsilon, and introduce trace and hermitian matrices, and some properties of rotations and reflections.
• Lectures 16-17 are about using matrices to solve linear equations.
• Lectures 18-20 are about eigenvalue and eigenvector stuff and change of basis.
• Lectures 21-22 are about quadratic forms.
• Lecture 23 is about Jordan normal form, in which I provide a proof in my notes.
• Lecture 24 is about uh some uh minkowski lorentz thing that can be applied to physics idk.

See my vectors and matrices notes here