Exercise:
Let Ain M_mtimes nK. Then textcol-rankAtextrow-rankA.

Solution:
Preparation for the proof bf Lemma . Let T:Vlongrightarrow W be a linear map and S:Wlongrightarrow L:Plongrightarrow V isomorphisms. Then: itemize item textrankScirc TtextrankT. item textrankTcirc LtextrankT. itemize isomorphisms don't change rank Proof of Lemma . itemize item textrankScirc TtextdimSTVtextdimTVtextrankT because S|_TV is an isomorphism between TV and STV. item textrankTcirc LtextdimTLPtextdimTV because LPV. itemize bf Lemma . Let Ain M_ntimes nK. Then A is invertible iff T_A:K^nlongrightarrow K^n is an isomorphism. Proof of Lemma Longrightarrow. Suppose A is invertible. We have T_A^-T_A^-. Proof of Lemma Longleftarrow. Suppose T_A is an isomorphism. We know that AT_epsilon_n^epsilon_n and by a previous result we obtain that A is invertible. bf Lemma . Let A Bin M_mtimes nK s.t. PAQ which means that they are equivalent where Pin textGL_mK Qin textGL_nK.Then: itemize item textcol-rankAtextcol-rankB. item textrow-rankAtextrow-rankB. itemize Proof of Lemma itemize item We have T_BT_Pcirc T_Acirc T_Q and both T_P T_Q are isomorphisms. By Lemma textrankT_BtextrankT_A. But textrankT_Btextcol-rankB textrankT_Atextcol-rankA Longrightarrow textcol-rankBtextcol-rankA. item We have B^TQ^T A^T P^T. Since P and Q are invertible so are also P^T Q^T we have seen P^T^-P^-^T By part of the Lemma we have textcol-rankB^Ttextcol-rankA^T textrow-rankBtextrow-rankA itemize Proof of the theorem. By a previous proposition exists Pin textGL_mK Qin textGL_nK s.t. PAQleftarray@c|c@ matrix I_r matrix & hline & matrix matrix arrayright denote this matrix by B. For B we have textcol-rankBtextrow-rankBr By Lemma textcol-rankAtextcol-rankBrquad textand textrow-rankAtextrow-rankBr &Longrightarrow textcol-rankAtextrow-rankA.