Vector spaces/Linear mapping/Homomorphism theorem/Surjective and kernel/Fact/Proof

Proof

The statement about existence and uniqueness follows from fact. We have to show that the uniquly determined group homomorphism $\tilde{\phi }$ is also compatible with scalar multiplication. Let $u\in Q$ be given with a preimage $v\in V$, and let $\lambda \in K$. Then $\lambda v$ is a preimage of $\lambda u$; therefore,

$\tilde{\phi }(\lambda u)=\phi (\lambda v)=\lambda \phi (v)=\lambda \tilde{\phi }(u),$

and $\tilde{\phi }$ is also compatible with the scalar multiplication.

To fact