Program, Semester Week 10

Linear Maps

When a transformation takes place between vector spaces, we call this transformation a map. Some maps of particular interest are linear maps, which we will study this week. You can think of a linear map as a function between vector spaces, where an input from one vector space is transformed - mapped - to an output in a possibly different vector space. For instance, calculating the magnitude of a vector is a map that as an input takes the vector and as an output produces a scalar.

A linear map gives rise to two interesting subspaces which tell a lot about the transformation that is taking place: the kernel, which contains all inputs that map to zero, and the image space, which is the set of all outputs. The rank-nullity theorem connects the dimensions of these spaces. By choosing ordered bases, a linear map can be represented by a mapping matrix which makes work with the map much easier. And it is in this context very useful to be able to change bases, which we will study a method for using a change-of-basis matrix.

Key Terms

Linear map \(L\). Mapping matrix \(\mathbf A\). Linear map expressed using a mapping matrix \(L_{\mathbf A}\). Kernel \(\mathrm{ker}\mathbf A\). Image space. The identity map and change-of-basis matrix \([\mathrm{id}]\). The rank-nullity theorem.

SymPy Demos

For this week we have SymPy demo 3 about linear maps.

Preparation and Syllabus

This week will cover Chapter [10 - Linear maps between vector spaces].

Exercises

Exercises for Long Day.

Exercises for Short Day.