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The following problem is an unsolved practice problem from a textbook which is assigned at the end of the chapter. So, I need hints on how to start solving the problem below:

Show that if $U$, $V$, $W$ are finite dimensional vector spaces, and $f\in \mathrm{Hom}\left ( U,V \right )$, $g\in \mathrm{Hom}\left ( V,W \right )$, then: $\dim \mathrm{Ker}\left ( gf \right )\leqslant \dim \mathrm{Ker}(f) + \dim \mathrm{Ker}(g)$

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    @m_p2009: That's not just information for me: that's information that is relevant to **all** readers. Please put it in the post, not the comments, and in the future please try to write in the context into the post from the very first.2012-01-20

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Note: my functions are written on the left of their argument.

Hint the first: $\mathrm{ker}(f)\subseteq \mathrm{ker}(g\circ f)$.

Hint the second: Let U' = \{ u\in U\mid f(u)\in\mathrm{ker}(g)\}. Now use the Rank-Nullity Theorem.

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    @m_p2009: as the last sentence of my last comment notes, $\mathrm{ker}(f|_{U'}) = \mathrm{ker}(f)\cap U' = \mathrm{ker}(f)$. So you have *equality* of subspaces, hence of dimensions.2012-01-20