# How to show that $V= \text{Im }f \oplus \text{Ker }g$.

Let $U, V, W$ be vector spaces over a field $K$, and let $f: U \to V$ and $g: V \to W$ be linear transformations such that $g \circ f$ is an isomorphism. I have to show that $V= \text{Im }f \oplus \text{Ker }g$.

I really don't know what to do, I've solved some similar problems but always with just one linear transfromation, not with two. I hope you can give me some hint or idea to handle this problem. Thank you in advance.

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Point one: show $\operatorname{Im} f \cap \ker g = \{0\}$. Point two: show every $v \in V$ can be written as $f(u) + k$, where $g(k) = 0$. –  Daniel Fischer Oct 13 '13 at 21:49
@DanielFischer: That sounds like an answer... –  Eric Stucky Oct 13 '13 at 22:01

$g\circ f\colon U\to W$; call $h\colon W\to U$ its inverse; then you have a way to go from $V$ back to $V$ with $\varphi=f\circ h\circ g$. If $v\in V$, consider $v=\varphi(v)+(v-\varphi(v))$ and go on.
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