Linear Algebra 2
Module code: MA1116
This module builds on the material studied in MA1115, with particular emphasis on the notions of abstract vector spaces and abstract linear maps. We will see that several things, such as matrices and vectors studied in MA1115, as well as sets of functions, can be viewed as vector spaces, and operations (including e.g. the notion of a derivative) on these sets act as linear maps; this is one of the reasons tools in linear algebra are applied in so many areas of mathematics. A core idea we shall study is the notion of the dimension of a vector space, in particular that of linear independence and bases. This idea will be used to unify the abstract notion of a vector space and linear map with the tools of matrices and vectors developed in the previous module.
The module will conclude with inner product spaces, an idea which allows for a general way to define lengths and angles in vector spaces, in particular studying special orthonormal bases of vector spaces. The topics studied in this module form a foundation for all areas of mathematics and its applications (in particular in e.g. analysis, differential equations, geometry, group theory, probability - and in sciences such as in AI and machine learning, cosmology, and particle physics).