= Reduced column echelon form
{title2=$\operatorname{rcef}(A)=\operatorname{rref}(A^T)^T$}
The reduced column echelon form of a <matrix> $A$ is defined by taking its <transpose>, forming the <reduced row echelon form>, and transposing back:
$$
\operatorname{rcef}(A)=\bigl(\operatorname{rref}(A^T)\bigr)^T.
$$
It is obtained from $A$ by <elementary column operations>. Its nonzero columns precede its zero columns, the topmost nonzero entry of each nonzero column is $1$, these leading entries move strictly downward as the column number increases, and each leading entry is the only nonzero entry in its row.
Every <elementary column operation> preserves the <column space>, because it and its inverse express each new column as a <linear combination> of the old columns. Thus the nonzero columns of the reduced form give a <basis> of the original <column space>. In contrast, <elementary row operations> preserve the <row space>. There is no general identity $\operatorname{rcef}(A)=\operatorname{rref}(A)^T$: the definition requires reducing $A^T$, not $A$.
For an <invertible matrix> of size $n$, both reduced forms are $I_n$. A singular example showing the distinction is
$$
A=\begin{pmatrix}1&2\\0&0\end{pmatrix},\qquad
\operatorname{rref}(A)=\begin{pmatrix}1&2\\0&0\end{pmatrix},\qquad
\operatorname{rcef}(A)=\begin{pmatrix}1&0\\0&0\end{pmatrix}.
$$
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