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https://texercises.com/exercise/bases-and-linear-maps/
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Exercise:
Let T:Vlongrightarrow W be a linear map between two finite dimensional vector spaces V and W. Let mathcalB mathcalC be bases for VW respectively. Then forall vin V: T^mathcalB_mathcalC v_mathcalBTv_mathcalC.

Solution:
Proof. The equality forall vin V: T^mathcalB_mathcalC v_mathcalBTv_mathcalC is equivalent to T_T^mathcalB_mathcalCPhi_mathcalCcirc Tcirc Phi_mathcalB^-. Since this is precisely how we defined the matrix T^mathcalB_mathcalC we've shown our claim.
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Exercise:
Let T:Vlongrightarrow W be a linear map between two finite dimensional vector spaces V and W. Let mathcalB mathcalC be bases for VW respectively. Then forall vin V: T^mathcalB_mathcalC v_mathcalBTv_mathcalC.

Solution:
Proof. The equality forall vin V: T^mathcalB_mathcalC v_mathcalBTv_mathcalC is equivalent to T_T^mathcalB_mathcalCPhi_mathcalCcirc Tcirc Phi_mathcalB^-. Since this is precisely how we defined the matrix T^mathcalB_mathcalC we've shown our claim.
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basis, eth, hs22, linear map, lineare algebra, proof, vector space
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(3, default)
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Language
ENG (English)
Type
Proof
Creator rk
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