universal property of the quotient space via homomorphiesatz
Proposition
Let 
be vector spaces and 
the kanonical projection.
Then precomposition induces a vectorspace-isomorphism
Warning
the proof consists purely of diagram chasing.
I think the proof injectivity is a bit absurd (albeit interesting)
There exist less abstract proofs :)
Proof
welldefined
We want to show that the map 
induced by precomposition restricts appropriately and that the RHS is a sub vector space.
sub vector space
restricts appropriately
omitted
bijective
setup
Let 
with 
.
Then in particular 
.
We will denote by 
the canonical projections to the quotient space
Furthermore we get a diagram
where 
is the vectorspace-isomorphism given by the second isomorphism
In particular we may identify the composition with 
injective - extra warning
Suppose 
with 
.
Then using our setup gives
in particular with 
where we use that 
is surjective.
Now the homomorphiesatz gives homomorphisms 
such that
commutes.
Under the canonical isomorphism 
by we get maps 
which are given as composition
In particular
commutes.
Now those maps are also maps making
commute, so uniqueness of the Homomorphiesatz shows 
.
Now
commutes, which shows that 
.
And at last we know that
commutes, so in particular 
what we wanted to show.
surjective
Now the Homomorphiesatz shows that there exists a unique map 
making the diagram commute
We claim that the composite 
is a preimage of .
This follows from
