Let $A$ be an $n \times p$ matrix with $n>p$. I'm wondering the following statement is true.
(i) $A$ has full rank
(ii) $Ax=0$ if and only if $x=0$
(iii) $Ay \neq 0$ for any $y \neq 0$
I know (i) and (ii) are equivalent, but not sure if (i) and (iii) are equivalent or not...
Since $A(0)=0,$ (ii) is equivalent to $$Ax=0\implies x=0.$$
Without any further knowledge in linear algebra, you should be certain that (ii) and (iii) are equivalent, because $P\implies Q$ and $\neg Q\implies\neg P$ are equivalent.