Given an exact sequence of vector spaces: $$0\longrightarrow U \longrightarrow V \longrightarrow W\longrightarrow 0$$ with $f:U\rightarrow V$ and $g: V \rightarrow W$
I want to prove the that following are equivalent:
$\bullet$ The sequence splits on the right ($\exists s:W\rightarrow V$ such that $g\circ s =1_W$)
$\bullet$ The sequence splits on the left ($\exists t: V\rightarrow U$ such that $t\circ f=1_U$)
$\bullet$ $\exists\gamma : V\rightarrow U\oplus W$ an isomorphism satisfying $\gamma \circ f=i_1$ and $p_2\circ \gamma=g$. Where $i_1$ and $p_2$ are the usual inclusion and projection into the first and second summand respectively.
So exactness gives us that $f$ is 1:1 and $g$ is onto, so clearly there exists a function with the property of the second bullet, but I can't quite figure out how to know it's linear on all of $V$ (or how this uses any assumptions from the first bullet.) I'm more confused about the latter two implications. Don't really know where to start with those unfortunately...
Edit: It says explicitly that I'm supposed to avoid using anything about bases here! For some reason...