The number of linearly independent solution of the homogeneous system of equations $AX=O$, where $X$ consists of $n$ unknowns and $A$ consist of $m$ linearly independent rows can't be equal to $n$
We know that for such a system the number of linearly independent solutions is given by $n-m$. When $n-m = n$ This implies $m=0$. That means number of linearly independent rows is $0$. So rank of $A$ should be $0$. And nullity is $n$. Is it true that if $A$ is a null matrix then there are $n$ linearly independent solution.
It seems like, all $n$ column vectors are solution for this system if $A$ is null matrix. And dimension of all $n$ column vectors is $n$. So the statement should be false. Am I correct$?$