about $\dim\operatorname{Ker}(F)$ and $\dim\operatorname{Im}(F)$

Question

Consider $F : \mathbb{R}^{23} \to \mathbb{R}^{20}$ such that its matrix $A$ has $rank(A)=17$. Find the dimensions $\dim\operatorname{Ker}(F)$ and $\dim\operatorname{Im}(F)$.

Any idea is welcome, merci !

Answer 1

Use the dimension theorem: if $\,T:V\to W\,$ is a linear transformation and $\,\dim V=n\,$, then $\,n=\dim\operatorname{Im}(T)+\dim\ker (T)\,$.

Remember: with the info and notation you gave, $\,\operatorname{rank}(A)=\dim\operatorname{Im}(T)$