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.

Key Terms

Span of vectors \(\mathrm{Span}\). Ordered basis for 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. Coordinates with respect to an ordered basis. The standard basis.

Preparation and Syllabus

This week will cover Sections 9.1 through 9.5 from Chapter 9 in [the textbook].

Exercises

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

Exercises for Short Day.