Homomorphic Image of Vector Space

Theorem
Let $\left({K, +_K, \times_K}\right)$ be a division ring.

Let $\left({V, +_V, \circ_V}\right)_K$ be a $K$-vector space.

Let $\left({W, +_W, \circ_W}\right)_K$ be a $K$-algebraic structure.

Let $\phi: V \to W$ be a homomorphism, i.e. a linear transformation.

Then the homomorphic image of $\phi$ is a $K$-vector space.

Proof
Let us write $\phi \left({V}\right)$ for the homomorphic image of $\phi$.

From Homomorphic Image of R-Module is R-Module, $\phi \left({V}\right)$ is a $K$-module.

It thus suffices to show that $\phi \left({V}\right)$ is unitary, since then it will be a $K$-vector space.

To this end, let $1_K$ be the unity of $K$.

Then for any $\phi \left({\mathbf v}\right) \in \phi \left({V}\right)$, compute:

Hence $\phi \left({V}\right)$ is unitary, and so a $K$-vector space.