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https://texercises.com/exercise/matrix-equivalence-and-rank-ii/
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Exercise:
Let AB in M_mtimes nK. Then Asim B iff textcol-rankAtextcol-rankB. Moreover forall leq rleq textminmn there is precisely one equivalence class of matrices A with textcol-rankr.

Solution:
Proof. Let ABin M_mtimes nK. Ase textcol-rankAtextcol-rankB and denote by r this common textcol-rank. By the previous corollar Asimleftarray@c|c@ matrix I_r matrix & hline & matrix matrix arrayright Bsimleftarray@c|c@ matrix I_r matrix & hline & matrix matrix arrayright. Since sim is an equivalence relation it follows that Asim B. Conversely ase Asim B Longrightarrow exists Pin textGL_mK Qin textGL_nK s.t. BPAQ Longrightarrow T_BT_Pcirc T_Acirc T_Q. textImT_BT_BK^nT_Pcirc T_AT_QK^nT_Pcirc T_AK^nT_PtextImT_A &Longrightarrow textdim ImT_BtextdimT_PtextImT_Atextdim ImT_A
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Exercise:
Let AB in M_mtimes nK. Then Asim B iff textcol-rankAtextcol-rankB. Moreover forall leq rleq textminmn there is precisely one equivalence class of matrices A with textcol-rankr.

Solution:
Proof. Let ABin M_mtimes nK. Ase textcol-rankAtextcol-rankB and denote by r this common textcol-rank. By the previous corollar Asimleftarray@c|c@ matrix I_r matrix & hline & matrix matrix arrayright Bsimleftarray@c|c@ matrix I_r matrix & hline & matrix matrix arrayright. Since sim is an equivalence relation it follows that Asim B. Conversely ase Asim B Longrightarrow exists Pin textGL_mK Qin textGL_nK s.t. BPAQ Longrightarrow T_BT_Pcirc T_Acirc T_Q. textImT_BT_BK^nT_Pcirc T_AT_QK^nT_Pcirc T_AK^nT_PtextImT_A &Longrightarrow textdim ImT_BtextdimT_PtextImT_Atextdim ImT_A
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equivalence, eth, hs22, lineare algebra, matrices, proof, rank, vector space
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