What's the difference between linear group action and group representation?

Question

Let $G$ be a group, and $V$ be a vector space. Let $\phi : G \times V \rightarrow V$ be a (left) linear group action of $G$ on $V$; and $\pi: G \rightarrow GL(V)$ be a representation of $G$ on $V$. Are they the same thing?

Answer 1

A group action on an object $X$ typically refers a homomorphism $G\to \operatorname{Aut}(X)$ for whatever the relevant category is: bijections for a set, homeomorphisms for a topological space, ring automorphisms for a ring, etc. (The first two are probably much more common.) A group action on the underlying set of $V$ that respects the vector-space structure of $V$ is exactly a representation $G\to GL(V)$.