System-Rank Theorem

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

(System-Rank Theorem). Let $[A|\vec{b}]$ be the augmented matrix of a system of $m$ linear equations in $n$ variables.

1. The system is consistent if and only if rank($A$) = rank ($[A\mid \vec{b}]$)
2. If the system is consistent, then the number of parameters in the general solution is the number of variables $n$ minus the rank of A:
   $\text{Number of parameters}=n-\text{rank}(A)$
3. The system is consistent for all $\vec{b}\in {\mathbb{R}}^m$ if and only if $\text{rank}(A)$= $m$.

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

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