–– Advanced Linear Algebra (Draft) ––

This is a draft of a textbook on advanced linear algebra, meaning it assumes some knowledge of vector spaces and linear maps. It is based on the second part of an introductory course given at a German university, and I am currently busy translating it into English. The idea is to adapt the textbook to fit the Australian curriculum, which requires reorganising the text somewhat, as an introductory section on algebraic structures such as groups and rings needs to be included. You can download the current draft here.

–– Chapter Overview (preliminary) ––

I Polynomials and Endomorphisms

Divisibility in Rings

• Units, Ideals and Divisibility
• Euclidean Rings
• Zeros of Polynomials
• Quotient Rings
• The Chinese Remainder Theorem

The Jordan Canonical Form

• Invariant Subspaces
• Nilpotent Endomorphisms
• The Programme
• The Primary Decomposition
• The Canonical Form of a Nilpotent Endomorphism
• The Jordan Canonical Form
• The Jordan Decomposition

II Applications: Codes and Ciphers

Cryptography

• Lagrange's Theorem
• Units in \(\mathbb{Z}/n\mathbb{Z}\)
• Diffie-Hellmann Key Exchange
• ElGamal Encryption

III Geometry in Vector Spaces

Vector Spaces with Inner Product

• The Canonical Inner Product

Appendix

The Geometry of Complex Numbers

Trigonometric Functions

–– Change Log ––

• 23.01.2014 uploaded draft, with chapters on divisibility in rings, Jordan canonical form and cryptography almost complete

–– To Do ––

• replace German references by English references
• write sections on real and general canonical forms