Agenda, Semester Week 8

Agenda, Semester Week 8#

Span, Ordered Basis and Coordinates#

This week we will once again work on vectors and matrices. Our first goal is to investigate a so-called span of a given set of vectors. We will introduce terms such as ordered basis and dimension for spans, as well as the terms kernel and column space of matrices. Furthermore, the rank-nullity theorem will give us a relation between these concepts. This will all lead us to the concept of coordinates for vectors in a span with respect to an ordered basis.

Remark: Python Test#

This Long Day the Python test for Python module 1 will take place. It will be based on the Jupyter Notebook released last week and will be carried out in Möbius, so bring your laptop. You must be present in the exercise area during the test (it cannot be done from home). It opens at 15:30 and closes at 17:00. All aids are allowed, also internet resources and AI tools.

Key Terms#

Long Day will cover:

  • Span of vectors \(\mathrm{Span}\) subsets of spans.

  • Ordered basis of a span.

  • Kernel \(\mathrm{ker}\) and column space \(\mathrm{colsp}\) of a matrix.

  • Dimension \(\mathrm{dim}\) and nullity \(\mathrm{null}\) of a matrix.

  • The rank-nullity theorem.

Short Day will cover:

  • Coordinates and coordinate vectors.

  • The standard basis.

Preparation and Syllabus#

Syllabus this week is Chapter 9.

  • For Long Day, read Chapter 9 through Section 9.4.

  • For Short Day, read Section 9.5.

Exercises#

Exercises for Long Day.

Exercises for Short Day.

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