row echelon form

Row Echelon Form

A matrix is in row echelon form if:
1] All zero rows are at the bottom
2] Each pivot is to the right of the one above it
3] All entries below a pivot are zero

$$\left[\begin{array}{cccc}\boxed{2}& 3& 1& 4\\ 0& \boxed{1}& 2& 3\\ 0& 0& 0& \boxed{5}\\ 0& 0& 0& 0\end{array}\right]$$

Reduced Row Echelon Form

A matrix in REF is in reduced row echelon form if additionally:
4] Each pivot is 1
5] Each pivot is the only non-zero entry in its column

$$\left[\begin{array}{cccc}\boxed{1}& 0& 1& 0\\ 0& \boxed{1}& 2& 0\\ 0& 0& 0& \boxed{1}\\ 0& 0& 0& 0\end{array}\right]$$

gaussian elimination transforms a matrix into REF (forward elimination) or RREF (Gauss-Jordan elimination).

The number of pivots equals the rank of the matrix.

Pivot columns vs free columns

When solving Ax = b:
Pivot columns correspond to basic variables, determined by back-substitution (solving from the bottom row upward, each pivot variable in terms of the ones already solved).
Non-pivot columns correspond to free variables (can be set to any value).

In the REF example above, columns 1, 2, 4 have pivots, column 3 doesn’t. So $x_3$ is free — the solution is parameterized by it.

Number of free variables $=n-\text{rank}(A)=\dim ($null space$)$.