Question: Does Theorem 1.9 need that (1, . . . , n) is a basis? That is, can we still get a well-defined and unique homomorphism

Does Theorem 1.9 need that (1, . . . , n) is a basis? That is, can we still get a well-defined and unique homomorphism if we drop either the condition that the set of 's be linearly independent, or the condition that it span the domain?

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