Row equivalent matrices have the same row rank

Prove that two row equivalent matrices over a field have the same row rank.


Let F be a field.

Suppose A and C are row equivalent via the matrix P, which is a product of elementary matrices. That is, PA = C, so that A^\mathsf{T} P^\mathsf{T} = C^\mathsf{T}. Let B and E denote the standard bases, and say A^\mathsf{T} = M^E_B(\varphi), C^\mathsf{T} = M^E_B(\psi), and P^\mathsf{T} = M^B_B(\theta). In particular, we have \varphi \circ \theta = \psi. By this previous exercise, the row rank of A is the column rank of A^\mathsf{T} is the dimension of \mathsf{im}\ \varphi, and the row rank of C is the column rank of C^\mathsf{T} is the dimension of \mathsf{im}\ \psi. Since \varphi \circ \theta = \psi and \theta is an isomorphism, we have \mathsf{im}\ \varphi \subseteq \mathsf{im}\ \psi, and likewise since \varphi = \psi \circ \theta^{-1}, \mathsf{im}\ \psi \subseteq \mathsf{im}\ \varphi. Thus we have \mathsf{im}\ \varphi = \mathsf{im}\ \psi, so that A and C have the same row rank.

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