# Theoretical linear algebra (Math 25A)

Undergraduate course, *Harvard University*, 2018

## Summary

In this course, we’ll pursue a rigorous treatment of linear algebra. Topics include set theory, vector spaces and bases, linear maps, determinants, eigenvectors, inner products, and spectral theory. The plan is to work through Axler’s Linear algebra done right, although we will use some other resources as well (see Course Materials below). This course is part one of a two-semester journey – the second semester will be real analysis and multivariable calculus.

The goals of the course include:

- Learning how to read and write proofs, and how to critique an argument. Learning to think carefully, logically, and rigorously. Learning to communicate math clearly and effectively.
- Learn both theoretical and computational aspects of linear algebra. More emphasis will be on the former, although both are important!
- Learn to use LaTeX. See the section on Homework below.
- Create a fun environment/community for learning and discussing math.

## Textbooks

- Simmons,
*Introduction to topology and modern analysis* - Alxer,
*Linear algebra done right* - Treil,
*Linear algebra done wrong*

## Topic schedule

- Week 1: Simmons Ch 1, sections 1-5
- Mon. Labor day (no class)
- Wed. Lecture 1: sets, functions, cardinality
- Fri.

- Week 2: Simmons Ch 1, sections 5-7
- Mon. Lecture 2: countability, equivalence relations
- Wed. Lecture 3: equivalence relations, more cardinality
- Fri. Homework 1 due.

- Week 3: Axler Ch 1
- Mon. Lecture 4: fields, vector spaces (definition, examples)
- Wed. Lecture 5: subspaces, direct sums, spanning sets
- Fri. Homework 2 due.

- Week 4 : Axler Ch 2
- Mon. Lecture 6: bases and dimension
- Wed. Lecture 7: linear independence theorem
- Fri. Homework 3 due.

- Week 5: Axler Ch 3, sections 3A-3B
- Mon. Lecture 8: linear maps, kernel/image
- Wed. Midterm 1.
- Fri. Extra credit due (Hilbert hotel, prime numbers, quotient spaces)

- Week 6: Axler Ch 3, section 3C
- Mon. Columbus day (no class).
- Wed. Lecture 9: matrices, rank-nullity
- Fri. Homework 4 due.

- Week 7: Axler Ch 3, section 3D; Treil Ch2
- Mon. Lecture 10: matrix multiplication, invertibility, linear systems
- Wed. Lecture 11: linear systems, row operations
- Fri. Homework 5 due.

- Week 8: Treil Chapters 2 and 3.
- Mon. Lecture 12: elementary matrices, inverses
- Wed. Lecture 13: determinants
- Fri. Homework 6 due.

- Week 9: Treil Chapter 3, 4; Axler Ch 5
- Mon. Lecture 14: more determinants
- Wed. Lecture 15: eigenvectors and polynomials
- Fri. Homework 7 due.

- Week 10: Treil Ch 4 (also to a lesser extent Axler Ch4)
- Mon. Lecture 16: polynomials, eigenvector existence, diagonalizability
- Wed. Lecture 17: more eigenvectors, diagonalizability
- Fri. Midterm 2. Extra credit due (exact sequences, error-correcting codes, tensor products, alternating forms)

- Week 11: Treil Ch 5 and Axler Ch 6
- Mon. Lecture 18: spectral theorem and inner products
- Wed. Lecture 19: inner products, orthogonality
- Fri. Homework 8 due.

- Week 12: Treil Ch 5 and Axler Ch 6
- Mon. Lecture 20: orthogonal complements, projections, and polynomial approximation
- Wed. Thanksgiving break (no class)
- Fri.

- Week 13: Treil Ch 6 and Axler Ch 7
- Mon. Lecture 21: dual spaces and adjoints
- Wed. Lecture 22: spectral theorem
- Fri. Homework 9 due.

- Week 14: Treil Ch 6 and Axler Ch 7
- Mon. Lecture 23: spectral theorem
- Wed. Lecture 24: spectral theorem. Homework 10 due.
- Fri. Extra credit due. (Google page-rank, universal property)