= Row-equivalent matrices
Two <matrices> of the same size over the same <field> are <row-equivalent matrices> if a sequence of <elementary row operations> transforms one into the other.<Interactive Linear Algebra by Dan Margalit and others/Row Reduction>[Interactive Linear Algebra by Dan Margalit and others; §1.2.1, definition of row equivalence]{ref} Every operation is reversible, so they have the same <row space> and the same column <linear relations>. Writing the entire sequence as multiplication by an <invertible matrix> gives $R=EA$ and
$$
Az=0\ \Longleftrightarrow\ Rz=0,\qquad
Ax=b\ \Longleftrightarrow\ Rx=Eb.
$$
Thus the <kernel> is preserved. For a nonzero right-hand side, perform every operation on the whole <augmented matrix> $[A\mid b]$: changing the coefficients while keeping $b$ fixed generally changes the solutions. The <column space> instead transforms to $E\operatorname{col}A$, which need not equal $\operatorname{col}A$.
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