Does every correspondence between bases, when extended to the spaces,

Does every correspondence between bases, when extended to the spaces, give an isomorphism? That is, suppose that V is a vector space with basis B = (1, . . . , n) and that f: B → W is a correspondence such that D = (f(1), . . . , f(n)) is basis for W. Must : V → W sending  = c11 + . . . + cnn to  () = c1(1) + ... + cn(n) be an isomorphism?

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