Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
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
up vote 0 down vote accepted


$g\circ f\colon U\to W$; 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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.