= Uniqueness of reduced row echelon form
Here is a short proof that uses only the <row space>. For a <vector subspace> $W\subseteq\mathbb F^n$, let $d_j$ be the <dimension> of its <projection map> onto the first $j$ coordinates, with $d_0=0$. In any <reduced row echelon form> with <row space> $W$, $d_j$ counts the <pivot columns> among those first $j$ columns: later-pivot rows project to zero, while earlier-pivot rows remain <linearly independent>. Thus the positions where $d_j-d_{j-1}=1$ determine all pivot positions from $W$ alone.
If those positions are $p_1,\ldots,p_r$, the map sending $w\in W$ to $(w_{p_1},\ldots,w_{p_r})$ is an <isomorphism> $W\to\mathbb F^r$: the reduced nonzero rows map to the <standard basis>. Its $i$th row is therefore the unique <vector> of $W$ with pivot-coordinate values equal to the $i$th <standard basis> vector. Any other <reduced row echelon form> with that <row space> has identical nonzero rows in identical order. At a fixed <matrix> size, the remaining rows must be zero. Together with existence by <Gaussian elimination>, this proves uniqueness.
Back to article page
