Definition:Linear Group Action

Definition
Let $(V,+,\cdot)$ be a vector space over a field $(k,\oplus,\circ)$.

Let $G$ be a group.

Let $\phi : G \times V \to V$ be an action of $G$ on $V$.

We say that $\phi$ is linear if it is compatible with the linear structure of $V$ in the following sense:


 * $ \forall v_1,v_2 \in V,\ g \in G : \phi(g,v_1+v_2) = \phi(g,v_1) + \phi(g,v_2) $


 * $ \forall \lambda \in k,\ g \in G,\  v \in V : \phi(g,\lambda \cdot v) = \lambda \cdot \phi(g,v) $

Also, $\left(V,\phi\right)$ is called a $G$-module.