# Part IA - Vectors and Matrices

## Lectured by N. Peake, Michaelmas 2014

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.

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# Contents

- V Full version
- 0 Introduction
- 1 Complex numbers
- 1.1 Basic properties
- 1.2 Complex exponential function
- 1.3 Roots of unity
- 1.4 Complex logarithm and power
- 1.5 De Moivre's theorem
- 1.6 Lines and circles in C
- 2 Vectors
- 2.1 Definition and basic properties
- 2.2 Scalar product
- 2.3 Cauchy-Schwarz inequality
- 2.4 Vector product
- 2.5 Scalar triple product
- 2.6 Spanning sets and bases
- 2.7 Vector subspaces
- 2.8 Suffix notation
- 2.9 Geometry
- 2.10 Vector equations
- 3 Linear maps
- 4 Matrices and linear equations
- 4.1 Simple example, 2 x 2
- 4.2 Inverse of an n x n matrix
- 4.3 Homogeneous and inhomogeneous equations
- 4.4 Matrix rank
- 4.5 Homogeneous problem Ax = 0
- 4.6 General solution of Ax = d
- 5 Eigenvalues and eigenvectors
- 5.1 Preliminaries and definitions
- 5.2 Linearly independent eigenvectors
- 5.3 Transformation matrices
- 5.4 Similar matrices
- 5.5 Diagonalizable matrices
- 5.6 Canonical (Jordan normal) form
- 5.7 Cayley-Hamilton Theorem
- 5.8 Eigenvalues and eigenvectors of a Hermitian matrix
- 6 Quadratic forms and conics
- 7 Transformation groups