Once a basis of V is chosen, linear maps f : V → W are completely determined by specifying the photographs of the premise vectors, because any aspect of V is expressed uniquely as a linear mixture of them. W, a 1-to-1 correspondence between fixed bases of V and W provides rise to a linear map that maps any foundation aspect of V to the corresponding foundation element of W. It is an isomorphism, by its very definition. Due to this fact, two vector spaces over a given field are isomorphic if their dimensions agree and vice versa. Another means
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