Agenda, Semester Week 6

Systems of Linear Equations

Linear equations appear very often throughout the natural sciences. Sometimes several linear equations appear simultaneously, often with much more than just one variable, and we have to find solutions that solve them all - we say that we are solving a system of linear equations. This week we will study how we can find all solutions to such systems and how the structure of such solutions looks. An important tool we will introduce is a so-called matrix along with its (reduced) row-echelon form.

Remark: Thematic Python Modules

On this week’s Long Day at 15:30 the first of a series of thematic Python modules will be released. It will be published via DTU Learn as a Jupyter Notebook file containing a set of Python-focused exercises. The purpose is to introduce the use of Python - in particular the Python package SymPy - for mathematical computations on the topics we have covered this far in the course. Next Long Day (in semester week 7, after the Autumn break) a test on this module will be carried out in Möbius. Details about this test will be found in the description of week 7.

Key Terms

Systems of linear equations. Matrices. Fields. The coefficient matrix and the augmented matrix. Gauss elimination and row operations. Reduced row-echelon form \(\mathrm{rref}\). Rank \(\rho\). Particular solutions and general solutions. The structural theorems.

Preparation and Syllabus

This week will cover Sections 6.1 through 6.4 from [the textbook].

Exercises

Exercises for Long Day. For the thematic Python module this Long Day a Jupyter Notebook file will be released at 15:30 on DTU Learn.

Exercises for Short Day.