Let L be the linear operator.

"Isomorphism --> Linear independence".

Isomorphism (for vector spaces) means a bijective linear operator. Bijective means both injective and surjective. Letbin R^n then there exists auniquexin R^n such that L(x)=b. That is, Ax=b. (Why?).

Thus, the linear system of equations,

Ax=bhas auniquesolution for every column vectorb.

Thus, det(A) != 0.

Thus, the column and row vectors in A are linearly independent.

Q.E.D.

See if you can do the converse.