Definition:Vector Space Monomorphism

Definition

Let $V$ and $W$ be $K$-vector spaces.

Then $\phi: V \to W$ is a vector space monomorphism if and only if:

$(1): \quad \phi$ is an injection

$(2): \quad \forall \mathbf x, \mathbf y \in V: \map \phi {\mathbf x + \mathbf y} = \map \phi {\mathbf x} + \map \phi {\mathbf y}$

$(3): \quad \forall \mathbf x \in V: \forall \lambda \in K: \map \phi {\lambda \mathbf x} = \lambda \map \phi {\mathbf x}$

Linguistic Note

The word monomorphism comes from the Greek morphe (μορφή) meaning form or structure, with the prefix mono- meaning single.

Thus monomorphism means single (similar) structure.