# If $\ T \$ is onto , then what is the dimension of the null space of $\ A \$

Suppose that $\ A \$ is a $\ 6 \times 13 \$ matrix and that $\ T(x)=Ax \$.

If $\ T \$ is onto , then what is the dimension of the null space of $\ A \$.

Since $\ A \$ is $\ 6 \times 13 \$ matrix , we can write $\ T : V \to W \$ ,

where $\dim(V)=13 \$ and $\ \dim (W)=6 \$

Now since $\ T \$ is onto map , $\ \operatorname{Im}(T)=W \$

Thus $\ \operatorname{rank}(T)=\dim (\operatorname{Im}(T))=\dim(W)=6 \$

Now ,

$\operatorname{rank}(T)+ \operatorname{Nullity}(T)=13 \ \Rightarrow \operatorname{Nullity}(T)=13-6=7 \$

Thus , $\operatorname{Nullity}(T)=7 \$