Suppose $V$ is a two-dimensional $K$-vector space. Does an embedding $\varphi: GL(V) \hookrightarrow GL(K^n)$ exist?
I tried multiplying $A \in GL(V)$ with the basis vectors of $K^n \cong \operatorname{Sym}^n(V)$ via $A(v\cdot w) = Av\cdot Aw$ and take the resulting coordinates as colums for $\varphi (A)$, but I don't see how this is well-defined (i.e. yields linearly independent colums and thus an invertible matrix) let alone a group homomorphism.
Edit: I forgot to mention that this embedding is in particular required to preserve the set of power tensors $v^n \in Sym^n(V)$ in a way that $A(v)^n = \varphi (A)(v^n)$ - So it is better to think of it as an embedding $\varphi: GL(V) \hookrightarrow GL(Sym^n(V))$.
I computed the above construction for $n = 3$ and it seemed to work, but I don't know why $\varphi$ needs to be a well-defined homomorphism in the general case.. Any hints would be greatly appreciated