the term "dimension" of a (finite dimension) vector space is well defined only after you prove that every basis for the vector space has the same size (and you can probably find the proof for this in any linear algebra book).

once you know that the dimension of V is n, then for a subset of size m<n, even if it is linearly independent, it cannot span the entire vector space because then it will be a basis of size m and there is also a basis of size n>m which is a contradiction.

if the subset is not independent, then you can take out vectors from it without changing the fact that it spans V until it is independent and then you got a new subset of size m'<m<n which is a basis and you again get a contradiction.