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https://texercises.com/exercise/ring-of-endomorphisms/
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Exercise:
Let V be a vector space over K. Then textEndV:textHomVV is a ring with a unity. This ring is in general not commutative. The multiplication product in textEndV is given by composition T_ T_ : T_circ T_ forall T_ T_in textHomVV. The unity is textid_V. Moreover if V is finite dimensional and textdimVn and mathcalBv_...v_n is basis for V then Psi_mathcalB^mathcalB: textHomVVlongrightarrow M_ntimes nK is an isomorphism of rings.

Solution:
Proof. The multiplication T_ T_ : T_circ T_ is associative T_ T_ T_T_circ T_circ T_T_circ T_circ T_T_ T_ T_. also T textid_V Tcirc textid_V T textid_V T textid_Vcirc T T. T T_+T_Tcirc T_+T_Tcirc T_ + Tcirc T_ etc. Now if V is finite dimensional and mathcalB is a basis for V then Psi_mathcalB^mathcalBT_circ T_T_circ T__mathcalB^mathcalBT__mathcalB^mathcalB T__mathcalB^mathcalB Psi_mathcalB^mathcalBT_ Psi_mathcalB^mathcalBT_.
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Exercise:
Let V be a vector space over K. Then textEndV:textHomVV is a ring with a unity. This ring is in general not commutative. The multiplication product in textEndV is given by composition T_ T_ : T_circ T_ forall T_ T_in textHomVV. The unity is textid_V. Moreover if V is finite dimensional and textdimVn and mathcalBv_...v_n is basis for V then Psi_mathcalB^mathcalB: textHomVVlongrightarrow M_ntimes nK is an isomorphism of rings.

Solution:
Proof. The multiplication T_ T_ : T_circ T_ is associative T_ T_ T_T_circ T_circ T_T_circ T_circ T_T_ T_ T_. also T textid_V Tcirc textid_V T textid_V T textid_Vcirc T T. T T_+T_Tcirc T_+T_Tcirc T_ + Tcirc T_ etc. Now if V is finite dimensional and mathcalB is a basis for V then Psi_mathcalB^mathcalBT_circ T_T_circ T__mathcalB^mathcalBT__mathcalB^mathcalB T__mathcalB^mathcalB Psi_mathcalB^mathcalBT_ Psi_mathcalB^mathcalBT_.
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endomorphism, eth, hs22, lineare algebra, proof
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(3, default)
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ENG (English)
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Proof
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