Agenda, Semester Week 10

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#

Long Day will cover:

  • Maps and linearity requirements.

  • Mapping matrices \([L]\).

  • Map compositions \(\circ\).

  • The identity map \(\mathrm{id}\).

  • Change-of-basis matrices \([\mathrm{id}]\).

Short Day will cover:

  • Image space of a linear map.

  • Kernel \(\mathrm{ker}\) of a linear map.

  • The rank-nullity theorem for a linear map.

Preparation and Syllabus#

Syllabus this week is Chapter 11.

  • For Long Day, read Chapter 11 through Section 11.3.

  • For Short Day, read Section 11.4.

Exercises#

Exercises for Long Day. A Python test for Python module 2 will take place today at 15:30-17:00.

Exercises for Short Day.

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