Let $G$ be a linear algebraic group over a field $K$ of characterstic zero acting on a vector space $V$. Then does this action induce a representation : $$\Gamma : Lie(G) \to gl(V)$$
If yes, how ? Please help me understand this. I would appreciate if the explanation is simple and from all prespectives like thinking of $Lie(G)$ as derivations on the coordinate ring or as the tangent space at identity of $G$.
A linear Group is also a linear map. Clearly linear maps can be represented as matrices. For a representation in $gl(V)$ it is required that no Matrix has vanishing determinant. This ensures in the case that $K$ has characteristic Zero that the linear Group is invertible. And Groups must be invertible per definition.
Therefore this representation exists.