Agenda, 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. These give rise to two interesting subspaces which tell a lot about the transformation that is taking place: the kernel and the image space. 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 using a change-of-basis matrix.

Key Terms

Linear maps. Mapping matrices. Kernel \(\mathrm{ker}\) and image space of a linear map. The identity map \(\mathrm{id}\) and change-of-basis matrices \([\mathrm{id}]\). The rank-nullity theorem for linear maps.

Preparation and Syllabus

This week will cover Sections 11.1 through 11.4 from Chapter 11 in [the textbook].

Exercises

Exercises for Long Day. A Jypyter Notebook for the third Python module will be released on Long Day at 15:30 on DTU Learn.

Exercises for Short Day.