Exercise 1.4.12

If the short wide matrix $A$ has $m<n$, we put $0$ as the extra column of $A$, then go with ‘elimination’. Since $m<n$, we only do $m-1$ eliminations, so there are going to be $n-(m-1)=n-m+1$ non zero elements at the last row (excluding the b=0 column). since $n-m+1>1$, this means that there are non-zero solution to the equations. The nullspace dimension is thus at least $n-m$.