Geometry for Programmers

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. ...