Yes. Whenever you have an "if and only if" statement, four implications follow from it:
- The forward direction, $P \implies Q$. "If the columns of $A$ are linearly independent, then the equation $Ax = 0$ has only the trivial solution."
- The contrapositive of the forward direction, $\neg Q \implies \neg P$. "If the equation $Ax = 0$ has a nontrivial solution, then the columns of $A$ are linearly dependent."
- The backward direction, $Q \implies P$. "If the equation $Ax=0$ has only the trivial solution, then the columns of $A$ are linearly independent."
- The contrapositive of the backward direction, $\neg P \implies \neg Q$. "If the columns of $A$ are linearly dependent, then the equation $Ax=0$ has a nontrivial solution."
Here, 1 and 2 are logically equivalent, as are 3 and 4. So if you were proving the if and only if statement, proving either one of 1 and 2, and then proving either one of 3 and 4, would be fine. If you're using the statement, you get to use whichever of the implications 1, 2, 3, 4 is relevant.