A finitely generated affine group scheme has a faithful finite-dimensional representation

Choose finitely many $k$-algebra generators $g_1,\dots ,g_m$ of $A$. By [F1] the coaction $\mathrm{\Delta }$ makes $A$ a comodule over itself, the regular representation, so by [F2] there is a finite-dimensional subcomodule $V\subseteq A$ containing $1,g_1,\dots ,g_m$. Since $\epsilon (1_A)=1_k\mathrm{\ne }0$, one has $1_A\mathrm{\ne }0$; hence $V\mathrm{\ne }0$ because it contains $1$.

Choose a basis $e_1,\dots ,e_n$ of $V$ and write $\mathrm{\Delta }(e_j)={\sum }_i{e}_i\otimes a_{ij}$ with $a_{ij}\in A$. By [F1] the coefficient identities $\mathrm{\Delta }(a_{ij})={\sum }_l{a}_{il}\otimes a_{lj}$ and $\epsilon (a_{ij})={\delta }_{ij}$ hold, and by the antipode identity of [F3] the matrix $(a_{ij})$ over $A$ has two-sided inverse $(S(a_{ij}))$, so $\det (a_{ij})$ is a unit. Hence $\mathrm{\Phi }\text{\hspace{0.17em}⁣}:k[x_{ij},d^{-1}]\to A$, $x_{ij}\mapsto a_{ij}$, $d^{-1}\mapsto \det (a_{ij}{)}^{-1}$, is a well-defined $k$-algebra homomorphism, and by [F1] it is the comorphism of the rational representation $r\text{\hspace{0.17em}⁣}:G\to {\mathrm{GL}}_n$ associated with the subcomodule $V\subseteq A$.

The image of $\mathrm{\Phi }$ contains every $a_{ij}=\mathrm{\Phi }(x_{ij})$, and the counit identity $(\epsilon \otimes \mathrm{id})\mathrm{\Delta }=\mathrm{id}$ gives $e_j={\sum }_i\epsilon (e_i)a_{ij}\in \mathrm{im}\mathrm{\Phi }$; hence $V\subseteq \mathrm{im}\mathrm{\Phi }$. Since $\mathrm{im}\mathrm{\Phi }$ is a $k$-subalgebra of $A$ containing $1$ and all generators $g_1,\dots ,g_m$ of $A$, it is all of $A$: $\mathrm{\Phi }$ is surjective.

By [F4] the morphism $\mathrm{Spec}\mathrm{\Phi }\text{\hspace{0.17em}⁣}:G\to {\mathrm{GL}}_n$ is a closed immersion. It is a morphism of group schemes: on $R$-points it is the group homomorphism $r_R$ (with ${\mathrm{GL}}_n(R)$ the invertible matrices), and two $k$-morphisms of affine schemes are equal exactly when they induce the same maps on $R$-points for every commutative $k$-algebra $R$, because the functor of points is fully faithful by the Yoneda lemma (The Yoneda bijection $\mathrm{Nat}(\mathcal{C}(a,-),F)\cong F(a)$ is natural in both $a$ and $F$, Affine schemes are contravariantly equivalent to commutative rings). Applying this to the morphisms $m_{{\mathrm{GL}}_n}\circ (r\times r)$ and $r\circ m_G$ and to the unit and inverse identities yields the three defining identities of a group-scheme morphism (Morphisms and closed subgroup schemes of group schemes). The representation is faithful: surjectivity of $\mathrm{\Phi }$ makes $g\mapsto g\circ \mathrm{\Phi }$ injective on $R$-points for every commutative $k$-algebra $R$, so each $r_R$ is injective. Choosing a basis identifies ${\mathrm{GL}}_V$ with ${\mathrm{GL}}_n$ and realizes the representation on the nonzero finite-dimensional space $V$, a subrepresentation of the regular representation.

If $G$ is any affine group scheme of finite type over $k$, then $A=\mathcal{O}(G)$ is a finitely generated $k$-algebra by [F5], using the assumed AC; steps 1.1-4.1 apply verbatim and produce the faithful finite-dimensional representation. The algebraic construction from a finitely generated Hopf algebra is choice-free: the coefficient calculation is finite, [F4] is choice-free, and [F2] uses only finite tensor expressions. AC is used to regard the constructed affine group objects, including ${\mathrm{GL}}_n$, as finite-type schemes via affine quasi-compactness.