I am just learning about action of $G$, algebraic group over an algebraically closed field $k$, that is locally algebraic. It states that if $G$ acts linearly on a vector space $W$, we say the action is locally algebraic if it is locally fintie and for any finite dimensional $G$-stable subspace $V$, the action $\theta: G \times V \rightarrow V$ is a morphism.
1) My first question is what is meant by " $G$ acts linearly on $W$"?
2) What is meant by " the action $\theta: G \times V \rightarrow V$ is a morphism"? Does this mean a morphism as algebraic groups or varieties?
3) Could someone explain why if $G\times V \rightarrow V$ is given by $$ (g, \sum_{i=1}^n \lambda_i e_i) \rightarrow \sum_{i} \lambda_i h_i(g^{-1})e_i $$ where $\lambda_i \in k$, $\{e_i\}$ is a basis of $V$ and $h_i \in k[G]$ then this defines a morphism?
1) $G$ acts linearly on $W$ means that each element of $G$ acts by a linear transformation of $W$. Equivalently the action determines a homomorphism of $G$ into $GL(W)$.
2) This means a morphism of varieties, the algebraic group structure of $V$ isn't relevant here.
3) At the level of (closed) points, a morphism of varieties is a map that's locally given by polynomials in affine charts. In this case, and $h_i(g^{-1})$ are polynomial in $g$, since inversion is a morphism on $G$, and $h_i\in k[G]$, so this map is polynomial in any affine chart of $G\times V$.