row equivalence

Two matrices are considered row equivalent if one can be changed to the other by a sequence of elementary row operations. This is most commonly applied to matrices that represent systems of linear equations, in which case two matrices of the same size are row equivalent if and only if the corresponding equations have the same set of solutions.

An example of such an equivalence would be \[\begin{pmatrix} 1&0&\bigm|&0\\-3&1&\bigm|&0 \end{pmatrix}\] and \[\begin{pmatrix} 1&0&\bigm|&0\\0&1&\bigm|&0 \end{pmatrix}\], which is obtain by doing \[R_{2}+3R_{1}\to R_{2}\].

The above corresponding system of homogenous equations are \[x=0,-3x+y=0\] and \[x=0,y=0\], which have the same set of solutions.