Prove that $\dim L(F)+\dim\ker(L)=\dim(F+\ker(L))$ for every subspace $F$ and every linear transformation $L$ of finite dimensional vector space $V$

Prove that $$\dim L(F) + \dim \ker(L)=\dim(F+\ker(L))$$ for every subspace $$F$$ and every linear transformation $$L$$ of finite dimensional vector space $$V$$.

Any idea would be appreciated.

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• Have you already learned the rank-nullity theorem? Because it is an immediate consequence of it. – Andreas Caranti Aug 1 at 10:03
• I know rank-nullity theorem but i dont know how to use it to solve this problem. – John Smith Aug 1 at 10:12
• Please see my answer below. – Andreas Caranti Aug 1 at 10:22

Apply rank-nullity to the linear map induced by $$L$$ from $$F + \ker(L)$$ to $$L(F)$$.
Consider the short exact sequence $$0\longrightarrow \ker L\longrightarrow F+\ker L\longrightarrow (F+\ker L)/\ker L\longrightarrow 0$$ and use the second isomorphism theorem: $$\;(F+\ker L)/\ker L\simeq F/(F\cap\ker L)$$, and observe that $$\;F/(F\cap \ker L)\simeq L(F)$$ by the first isomorphism theorem.