This site is devoted to mathematics and its applications. Created and run by Peter Saveliev.

# Linear algebra: course

### From Mathematics Is A Science

## 1 Description

This is a one-semester course in linear algebra and vector spaces. An emphasis is made on the coordinate free analysis. The course mimics in some ways a modern algebra course.

## 2 Prerequisites

## 3 Lectures

- Vector spaces: introduction
- More on vector spaces
- Solving systems of linear equations
- Internal structure of a vector space: part 1
- Internal structure of a vector space: part 2
- Internal structure of a vector space: part 3
- Matrices: part 1
- Matrices: part 2
- Matrices as functions
- Linear operators: part 1
- Linear operators: part 2
- Linear operators: part 3
- Linear operators: part 4
- Linear operators: part 5
- Determinants of linear operators
- Eigenvalues and eigenvectors of linear operators
- Dual spaces
- Diagonalization of matrices
- Inner product spaces: part 1
- Vector space of infinite sequences
- Inner product spaces: part 2

## 4 Exercises and tests

- Linear algebra: exercises
- Linear algebra: homework 1
- Differential forms: homework 1
- Linear algebra: test 1
- Linear algebra: test 2
- Linear algebra: final
- Review exercises

## 5 Notes

The content came from this complete set of handwritten lectures.

Texts:

The following topics weren't addressed enough:

- quotients, products, etc,
- infinite-dimensional spaces,
- inner products spaces,
- dual spaces,
- eigenvalues and eigenvectors,

or not at all:

- multilinear algebra;
- tensor product;
- vector spaces over general fields, modules.

All are required for serious applications of linear algebra.