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**Isomorphism** You are mixing terms here. A mere set does not have the luxury of a dimension. A set thats a Vector Space, has dimensions.

So there is no meaning attached to the term, dimension of set S. However if S is a set of vectors, then the term dimension of span of S is perfectly meaningful.

If you define the dimension of a vector space to be the number of elements in the basis of that vector space and define the basis as the minimal spanning set for that vector space, you just have to prove that span(S) is a minimal spanning set. But thats obvious, because if you drop any vector from S, then the remaining vectors cant combine to give the dropped vector(why?). Thus the set is a minimal spanning set.

Hence S is a basis for span(S). But S has m vectors and thus dim(span(S)) = m.