Agenda, Semester Week 6

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.

Key Terms#

Long Day will cover:

  • Systems of linear equations.

  • Matrices.

  • Fields \(\mathbb F\).

  • Coefficient matrix and augmented matrix.

  • Gauss elimination and row operations.

  • Reduced row-echelon form \(\mathrm{rref}\).

Short Day will cover:

  • Rank \(\rho\).

  • Free parameter.

  • Particular solutions and general solutions.

  • The structural theorems.

Preparation and Syllabus#

Syllabus this week is Chapter 6 - note that its final section is optional.

  • For Long Day, read Chapter 6 through Section 6.3.

  • For Short Day, read Section 6.4.

Exercises#

Exercises for Long Day.

Exercises for Short Day.

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