splitting Lemma for vector spaces
Proposition
Let 
be a field, 
vector spaces and
be a short exact sequence, i.e.
(here the second and third equation precisely state, that 
is injective and 
is surjective)
Then there exists a vectorspace-isomorphism 
such that the diagram
here
are the evident vectorspace homomorphisms.
Proof
we construct a morphism
show that 
is a bijection (hence an isomorphism) and that the diagrams commute.
morphism
choose a basis 
of 
.
Then by assumption is injective, hence 
is linearly independent (cf. Vektorraummonomorphismus erhält lineare Unabhängigkeit).
We may extend 
to a basis (cf. erweiterung einer linear unabhängigen Menge zu einer Basis) 
and define
where the map 
is welldefined (existence follows from assumption, uniqueness follows from injectivity of ).
It follows that
for example since 
(cf. Existenz und Eindeutigkeit eines Homomorphismus durch die Abbildungen der Basis)
Then define
bijective
injective
Let 
.
Then in particular 
and we may find a preimage 
of 
under (since 
.
It follows that
which shows 
or 
.
Thus we are done by minimal kernel and monomorphism
surjective
by hand for finite dimensional 
general case
Let 
.
Then choose a one side inverses 
with
(cf. vector space epimorphism splits, vector space monomorphism splits)
Then define
Then applying shows
It remains to show that
Here we get
and
since 
commutes
follows as
since 
and
