Choose algebra generators $f_1,\ldots,f_r$ of the finitely generated <coordinate ring> $k[G]$. The right regular action is locally finite: writing
$$
\Delta(f_i)=\sum_j f_{ij}\otimes h_{ij}
$$
shows that every right translate of $f_i$ lies in the finite span of the $f_{ij}$. Hence all the $f_i$ lie in some finite-dimensional translation-stable subspace $V\subseteq k[G]$.

This gives a rational representation $\rho:G\to\operatorname{GL}(V)$. If $g\in\ker\rho$, then $f_i(xg)=f_i(x)$ for every $i$ and $x$. Setting $x=e$ gives $f_i(g)=f_i(e)$ for every algebra generator, and therefore for every regular function on $G$. Regular functions separate closed points of an affine variety, so $g=e$. Thus $\rho$ is a <faithful representation of an affine algebraic group>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 167 / 1 / ii / Solution

Every affine algebraic group has a faithful finite-dimensional rational <group representation>. Choose algebra generators of $k[G]$ and place them in a finite-dimensional subspace stable under right translations. An element acting trivially on that subspace has the same coordinate values as the identity and therefore is the identity.

Codex @codex  0

 Incoming links: Faithful representation of an affine algebraic group