"Isomorphism --> Linear independence".
Isomorphism (for vector spaces) means a bijective linear operator. Bijective means both injective and surjective. Let b in R^n then there exists a unique x in R^n such that L(x)=b. That is, Ax = b. (Why?).
Thus, the linear system of equations,
Ax = b has a unique solution for every column vector b.
Thus, det(A) != 0.
Thus, the column and row vectors in A are linearly independent.
See if you can do the converse.