Definition:Linear Group Action

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Definition

Let $\left({V, +, \cdot}\right)$ be a vector space over a field $\left({k, \oplus, \circ}\right)$.

Let $G$ be a group.

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

Then $\phi$ is a (left) linear group action if and only if it is compatible with the linear structure of $V$ in the following sense:

$(1): \quad \forall v_1, v_2 \in V: g \in G: \phi \left({g, v_1 + v_2}\right) = \phi \left({g, v_1}\right) + \phi \left({g, v_2}\right)$
$(2): \quad \forall \lambda \in k, g \in G, v \in V: \phi \left({g, \lambda \cdot v}\right) = \lambda \cdot \phi \left({g, v}\right)$