Homomorphic Image of Vector Space

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Theorem

Let $\struct {K, +_K, \times_K}$ be a division ring.

Let $\struct {V, +_V, \circ_V}_K$ be a $K$-vector space.

Let $\struct {W, +_W, \circ_W}_K$ 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 $\phi \sqbrk V$ for the homomorphic image of $\phi$.

From Homomorphic Image of R-Module is R-Module, $\phi \sqbrk V$ is a $K$-module.

It thus suffices to show that $\phi \sqbrk V$ 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 $\map \phi {\mathbf v} \in \phi \sqbrk V$, compute:

\(\displaystyle 1_K \circ_W \map \phi {\mathbf v}\) \(=\) \(\displaystyle \map \phi {1_K \circ_V \mathbf v}\) $\phi$ is a linear transformation
\(\displaystyle \) \(=\) \(\displaystyle \map \phi {\mathbf v}\) $V$ is a $K$-vector space


Hence $\phi \sqbrk V$ is unitary, and so a $K$-vector space.

$\blacksquare$


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