Nullity and Rank of Matrix

Question

Given $A$ is an $m\times n$ matrix, where $m<n$. Since $m<n$ then I can conclude that the reduced form $A$ will have free variables, hence $dim$ $nullA>0$. Let's say $dim$ $nullA=1$. Since $m<n$, therefore $n$ must not be less than 2 and we have $dim$ $nullA$ $+$ $dim$ $colA=n$, does this implies that $dim$ $colA>0$ aswell?

Answer 1

Since $1 \le m<n$ , we have $ \dim col A =n- \dim null A = n-1 >0$.