# Linear Algebra: Vector Spaces

These notes take the Linear Algebra (First Year) and develop them to abstract vector spaces. These are defined by means of a set of axioms, generalising the concept of vector. Linear transformations are developed in this context and are shown to be intimately related to the matrices we have been working with all along. Some of the most important examples of vector spaces are spaces of functions, such as the space of solutions to some differential equation.

Inner product spaces are vector spaces with some extra structure that enable us to define lengths and angles. These concepts are useful in geometric spaces, where orthogonality is equivalent to perpendicularity if the vectors are non-zero. But their more important use is in spaces of functions, especially in an area of mathematics called Fourier Theory.

[Please note that all links are to Adobe .pdf documents. They will open in a separate browser window.]

- Introduction and Contents
- CHAP01 Vector Spaces
- CHAP02 Linear Transformations
- CHAP03 Inner Product Spaces
- CHAP04 Diagonalisation Revisited
- CHAP05 Conics and Quadric Surfaces
- CHAP06 Jordan Canonical Form
- CHAP07 Numerical Linear Algebra