# System-Rank Theorem

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

**(System-Rank Theorem)**. Let $[A∣b]$ be the augmented matrix of a system of $m$ linear equations in $n$ variables.

- The system is consistent if and only if rank($A$) = rank ($[A∣b]$)
- 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:

$Number of parameters=n−rank(A)$ - The system is consistent for all $b∈R_{m}$ if and only if $rank(A)$= $m$.

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.