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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 !

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Rank-nullity theorem is your friend. – Jyrki Lahtonen Dec 4 '12 at 11:20
up vote 4 down vote accepted

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, $\,\operatorname{rank}(A)=\dim\operatorname{Im}(T)$

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