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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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  • $\begingroup$ Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be put on hold. To prevent that, please edit the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers. $\endgroup$ – José Carlos Santos Aug 1 at 10:01
  • $\begingroup$ Have you already learned the rank-nullity theorem? Because it is an immediate consequence of it. $\endgroup$ – Andreas Caranti Aug 1 at 10:03
  • $\begingroup$ I know rank-nullity theorem but i dont know how to use it to solve this problem. $\endgroup$ – John Smith Aug 1 at 10:12
  • $\begingroup$ Please see my answer below. $\endgroup$ – Andreas Caranti Aug 1 at 10:22
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Hint

Apply rank-nullity to the linear map induced by $L$ from $F + \ker(L)$ to $L(F)$.

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  • $\begingroup$ Thank you very much sir.It`s really helpful. $\endgroup$ – John Smith Aug 1 at 11:13
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Hint:

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.

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