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https://texercises.com/exercise/inverses-and-isomorphisms/
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Exercise:
Let Ain M_ntimes nK. Then Ain textGL_nK i.e. A is invertible iff T_A:K^nlongrightarrow K^n is an isomorphism. Moreover T_A^-T_A^-.

Solution:
Proof. Ase A is invertible. We claim that T_A^-circ T_Aid_K^n and T_Acirc T_A^-id_K^n which will show that T_A is an isomorphism. Indeed forall vin K^n:T_A^-circ T_AvA^- T_AvA^- A v I_n v v. Similarly one shows that forall vin K^n:T_Acirc T_A^-vI_n v v. This proves that T_A is an isomorphism. Conversely ase T_A:K^nlongrightarrow K^n is an isomorphism. Let S: T_A^-. By a previous result ST_B for some Bin M_ntimes nK. Now forall vin K^n: vScirc T_AvT_BA vB A v B A v We apply this equality for ve_ ve_...v_e_n and obtain that B AI_n. Similarly one shows that A BI_N too Longrightarrow A is invertible.
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Exercise:
Let Ain M_ntimes nK. Then Ain textGL_nK i.e. A is invertible iff T_A:K^nlongrightarrow K^n is an isomorphism. Moreover T_A^-T_A^-.

Solution:
Proof. Ase A is invertible. We claim that T_A^-circ T_Aid_K^n and T_Acirc T_A^-id_K^n which will show that T_A is an isomorphism. Indeed forall vin K^n:T_A^-circ T_AvA^- T_AvA^- A v I_n v v. Similarly one shows that forall vin K^n:T_Acirc T_A^-vI_n v v. This proves that T_A is an isomorphism. Conversely ase T_A:K^nlongrightarrow K^n is an isomorphism. Let S: T_A^-. By a previous result ST_B for some Bin M_ntimes nK. Now forall vin K^n: vScirc T_AvT_BA vB A v B A v We apply this equality for ve_ ve_...v_e_n and obtain that B AI_n. Similarly one shows that A BI_N too Longrightarrow A is invertible.
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eth, hs22, inverse, isomorphism, lineare algebra, matrices, proof, vector space
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ENG (English)
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Proof
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