System-Rank Theorem

This is super important, basis of understanding how many solutions a system of linear equations has.

(System-Rank Theorem). Let be the augmented matrix of a system of linear equations in variables.

  1. The system is consistent if and only if rank() = rank ()
  2. If the system is consistent, then the number of parameters in the general solution is the number of variables minus the rank of A:
  3. The system is consistent for all if and only if = .

Note: (A alone with just the coefficients is called the coefficient matrix)

Definition: A linear system of equations in variables is underdetermined if , this is, if it has more variables than equations.