A system of linear equation is a bunch of equations where the left hand side is a linear function and the right hand side is a constant.

A linear system can be written in the format

where we need to solve .

Visualization

For any linear system, there are some good mental models to visualize:

linear_equations.png

Solution

Not all linear systems can be solved. A system of linear equations has either

  1. no solution, or,
  2. exactly one solution
  3. an infinite number of solutions

Whether the system of linear equations have a solution depends on the numbers of equations versus the number of unknowns. For a system of linear equation where

  • (more equations than unknowns), we expect no solution
  • (less equations than unknowns), we expect many solutions
  • , one solution if is invertible

Solvers

There are two types of linear solvers: direct and iterative.

A direct solvers find the exact solution in a finite number of steps, without relying on iterative approximation. Gaussian elimination, LU factorization, and Cramer’s rule are all considered direct methods. 1 2

Footnotes

  1. Direct Methods for Solving Linear Systems of Equations

  2. ljll.fr/~frey/cours/UPMC/linear-systems.pdf