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https://texercises.com/exercise/matrix-equivalence-and-rank/
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Exercise:
Let Ain M_mtimes nK. Then exists Pin textGL_mK Qin textGL_nK s.t. PAQ leftarray@c|c@ matrix I_r matrix & hline & matrix matrix arrayright where rtextcol-rankA. In particular textcol-rankAleq textminmn.

Solution:
Proof. Consider T_A:K^nlongrightarrow K^m T_AvA v. Then rtextcol-rankAtextdim ImT_A because we have seen that textImT_AtextSpT_Ae_...Te_ntextSptextcolumns of A. Now apply the previous proposition to T_A and we get that T_mathcalC^mathcalB leftarray@c|c@ matrix I_r matrix & hline & matrix matrix arrayright for some bases mathcalB of K^n and mathcalC of K^m. &Longrightarrow id_K^m_mathcalC^epsilon_m T_A_epsilon_m^epsilon_n id_K^n_epsilon_m^mathcalB leftarray@c|c@ matrix I_r matrix & hline & matrix matrix arrayright where epsilon_n epsilon_m are the standard bases of K^n K^m respectively.
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Exercise:
Let Ain M_mtimes nK. Then exists Pin textGL_mK Qin textGL_nK s.t. PAQ leftarray@c|c@ matrix I_r matrix & hline & matrix matrix arrayright where rtextcol-rankA. In particular textcol-rankAleq textminmn.

Solution:
Proof. Consider T_A:K^nlongrightarrow K^m T_AvA v. Then rtextcol-rankAtextdim ImT_A because we have seen that textImT_AtextSpT_Ae_...Te_ntextSptextcolumns of A. Now apply the previous proposition to T_A and we get that T_mathcalC^mathcalB leftarray@c|c@ matrix I_r matrix & hline & matrix matrix arrayright for some bases mathcalB of K^n and mathcalC of K^m. &Longrightarrow id_K^m_mathcalC^epsilon_m T_A_epsilon_m^epsilon_n id_K^n_epsilon_m^mathcalB leftarray@c|c@ matrix I_r matrix & hline & matrix matrix arrayright where epsilon_n epsilon_m are the standard bases of K^n K^m respectively.
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equivalence, eth, hs22, lineare algebra, matrices, proof, rank
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ENG (English)
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Proof
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