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 iff $\text{rank(A)}=\text{rank}([A|\vec{b}])$
2. If the system is consistent then: $\text{num. of free params}=n-\text{rank(A)}$
3. The system is consistent for all $\vec{b}\in R^m$ iff $\text{rank(A) = m}$

Reason for 1

• If we have a bad row (say bottom row).
• That row in $A$ will not have a pivot.
• But the same row, when considering $[A|\vec{b}]$, will have a pivot as the row still has a constant term on RHS.

$$⟹\text{rank(A)}\ne \text{rank}([A|\vec{b}])$$

Reason for 2

$n$ is the number of columns.

• We only have a free parameter if a column does not have a pivot.
• If each column has a pivot, then we would have the same number of pivots as columns.

Reason for 3

• If our system is consistent, we have no bad rows.
• I.e. each of our rows has a pivot.