chapter three

3 The geometry of linear equations

 

This chapter covers

  • Learning the geometrical sense of systems of linear equations
  • Telling which systems could possibly be solved
  • Understanding iterative solvers, including convergence, stability, and exit condition
  • Understanding direct solvers and algorithmic complexity
  • Picking the best solver for any particular system

Systems of linear equations are everywhere. In fact, we solved one in the first chapter. Remember the two-trains problem?

solution = solve([
    Vp - Va * 2,
    Va * 1 + Vp * 1 - 450
], (Va, Vp))

Yes, this is a small system of linear equations. It has two equations, which makes it a system, and each equation is a sum of scaled variables supplied optionally with a number, which makes equations linear.

3.1 Linear equations as lines and planes

3.1.1 Introducing a hyperplane

3.1.2 A solution is where hyperplanes intersect

3.1.3 Section 3.1 summary

3.2 Overspecified and underspecified systems

3.2.1 Overspecified systems

3.2.2 Underspecified systems

3.2.3 Section 3.2 summary

3.3 A visual example of an interactive linear solver

3.3.1 The basic principle of iteration

3.3.2 Starting point and exit conditions

3.3.3 Convergence and stability

3.3.4 Section 3.3 summary

3.4 Direct solver