= Reduced row echelon form
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A <row echelon form> is in <reduced row echelon form> when every <pivot element> is one and is the only nonzero entry in its column.<Interactive Linear Algebra by Dan Margalit and others/Row Reduction>[Interactive Linear Algebra by Dan Margalit and others; §1.2.2, reduced echelon conditions 4 and 5]{ref} This extra reduction displays each pivot unknown directly in terms of the <free variables in a linear system>. The zero <matrix> also satisfies the definition: it has no <pivot elements>.
Every <matrix> has exactly one <reduced row echelon form> under <elementary row operations>, with its size and column order held fixed.<Interactive Linear Algebra by Dan Margalit and others/Row Reduction>[Interactive Linear Algebra by Dan Margalit and others; §1.2.3, theorem preceding the Row Reduction Algorithm]{ref} Ordinary <row echelon form> is not unique: multiplying a nonzero row by two over the <rational numbers> keeps the staircase but changes the <matrix>. The reduced conditions remove that freedom.
For a fixed number of columns, each <row space> has a unique reduced <matrix> after zero rows are removed; at a fixed full size, the number of appended zero rows is also determined. Thus <reduced row echelon form> represents a <vector subspace> independently of the starting <basis>. This is the uniqueness used by <Gaussian binomial coefficients> to count <vector subspaces> over a <finite field>.
The fixed column order matters. <Elementary column operations> can change the <row space>, the <kernel> and the resulting <reduced row echelon form>. For instance, swapping the columns of $(1\ \ 2)$ over the <rational numbers> gives $(2\ \ 1)$, whose <reduced row echelon form> is $(1\ \ 1/2)$ instead of $(1\ \ 2)$. When solving a <linear system>, a column interchange also requires tracking the corresponding interchange of unknowns.
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