This is a proposition from Artin's Algebra (1st edition, Proposition 5.12):
The presentation matrix $A$ of an $R$-module is the same as deleting a column of zeros of $A$.
He proves it by stating that a column of zeros corresponds to the trivial relation which can be deleted.
I am quite confused.
1) To talk about relation, we must first have a set of elements in $V$, the relation here refers to what set of elements in $V$? My guess is the generating set of $V$, but in his theorem he is talking about a general module $V$ which might not have a generating set.
2) Even though we confine ourselves to finitely generated $R$-module $V$, does $A$ always correspond to a complete relation of some generating set in $V$? I do not see why this must be true if we are only given an isomorphism between $R^m/AR^n$ and $V$.