Let $k$ be a field of characteristic zero and $G$ a reductive group over $k$. Furthermore, let $X$ be a projective $k$-variety with a $G$-action. Then we know, for example from Mumfords book about GIT, that there is an $n \in \mathbb{N}$ and a $G$-equivariant embedding $$\iota:X \hookrightarrow \mathbb{P}^n_k$$ where $G$ acts on $\mathbb{P}^n_k$ via a representation $\rho:G \rightarrow GL_{n+1}$.

Can we always assume that $\rho$ is faithful?

Let $k$ be a field of characteristic zero and $G$ a reductive group over $k$. Furthermore, let $X$ be a projective $k$-variety with a $G$-action. Then we know, for example from Mumfords book about GIT, that there is an $n \in \mathbb{N}$ and a $G$-equivariant embedding $$\iota:X \hookrightarrow \mathbb{P}^n_k$$ where $G$ acts on $\mathbb{P}^n_k$ via a representation $\rho:G \rightarrow \mathrm{GL}_{n+1}$.

Can we always assume that $\rho$ is faithful?