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Week 5, Long Day: Systems of Linear Equations

Today we will be working with systems of equations. We are already more than capable of solving single linear equations, but systems of them can quickly be a time consuming challenge without dedicated methods. The subject will be new to most students as we will introduce the concept of matrix notation, but it shouldn’t be feared; essentially we are still just dealing with the four elementary arithmetic operations! We will work out how so-called coefficient matrices and augmented matrices can represent systems of equations in a smarter way and we will see how this notation form allows for solving the equations efficiently via Gauss-Jordan elimination.

Matrices can also be studied in their own right and we will come to view them as entirely new mathematical objects that can be added together, multiplied by each other, etc. Then suddenly a system of linear equations can be viewed as a matrix equation $\,\mathbf A \mathbf x=\mathbf b\,$ where $\,\mathbf A\,$ is a known matrix, $\,\mathbf b\,$ a known vector (the right-hand-side), and $\,\mathbf x\,$ the unknown vector that we are looking to find.

Today’s Key Concepts

Multiple-dimensional number spaces, both real and complex. Systems of linear equations. Coefficient matrix. Augmented matrix. Gauss-Jordan elimination. General solution. Free parameters. Rank. Matrix-vector product and matrix-matrix product.

Preparation and Syllabus

Today’s material covers eNote 6 Systems of Linear Equations and most of eNote 7 Matrices and Matrix Algebra.

Maple Ressources

  • RowOperation performs a row operation.
  • ReducedRowEchelonForm reduces a system of equations to its reduced echelon form.
  • LinearSolve solves systems of linear equations.

Today two Maple demos are available for extra training: EquationSystems and MatrixAlgebra

Activity Program

  • 10.00 – 12.00: Lecture (aud. 42, b. 303A) (link to streaming)
  • 12.30 – 17.00: Group exercises in the study areas (b. 302, bottom floor)
  • 13.00 – 16.00: Your teachers are present in the study areas

Group Exercises:

  1. Solutions to Systems of Equations
  2. Intro to Systems of Equations with Maple
  3. Systems of Linear Equations with Maple
  4. The Structure of Solutions and the Concept of Rank: Theory
  5. The Structure of Solutions and the Concept of Rank: Practice
  6. Calculations with Matrices
  7. A System of Equations in disguise?