# Part IB - Linear Algebra

## Lectured by S. J. Wadsley, Michaelmas 2015

These notes are not endorsed by the lecturers, and I have modified them (often significantly) after lectures. They are nowhere near accurate representations of what was actually lectured, and in particular, all errors are almost surely mine.

This is an HTML version of the notes, generated using some horribly-written scripts and pdf2htmlEX. These are not guaranteed to display well, but do let me know if something is broken. Note however that I cannot help you if your browser does not support standard HTML features (eg. this part is known not to work well with w3m). You can either view all sections in a single page (Full version), or access individual sections below. If you want to download a pdf, head to the Notes page.

# Contents

- V Full version
- 0 Introduction
- 1 Vector spaces
- 1.1 Definitions and examples
- 1.2 Linear independence, bases and the Steinitz exchange lemma
- 1.3 Direct sums
- 2 Linear maps
- 2.1 Definitions and examples
- 2.2 Linear maps and matrices
- 2.3 The first isomorphism theorem and the rank-nullity theorem
- 2.4 Change of basis
- 2.5 Elementary matrix operations
- 3 Duality
- 4 Bilinear forms I
- 5 Determinants of matrices
- 6 Endomorphisms
- 6.1 Invariants
- 6.2 The minimal polynomial
- 6.3 The Cayley-Hamilton theorem
- 6.4 Multiplicities of eigenvalues and Jordan normal form
- 7 Bilinear forms II
- 8 Inner product spaces