I would like the find an embedding/faithful representation from the projective linear group $PGL_n\mathbb{C}\to GL_m\mathbb{C}$ for some $m$, and likewise for the other projective groups $PSL_n\mathbb{R}, PSO_n\mathbb{C}, PSO_n\mathbb{R}, Psp_{2n}\mathbb{C}$, and $Psp_{2n}\mathbb{R}$.
Is there an elementary way to deduce such an embedding, without using the machinery of Lie algebras or adjoint representations?
I know that we can make an argument using the adjoint representation of $PGL_n\mathbb{C}$ on its Lie algebra, but these haven't been introduced in my text yet.
A representation of $\text{PGL}_n(\mathbb{C})$ is, almost by definition, a representation of $\text{GL}_n(\mathbb{C})$ on which the center acts trivially. The center of a group $G$ is, almost by definition, the kernel of the homomorphism
$$G \ni g \mapsto (x \mapsto gxg^{-1}) \in \text{Aut}(G).$$
When $G = \text{GL}_n(\mathbb{C})$, this homomorphism extends to a linear representation of $\text{GL}_n(\mathbb{C})$ on $\mathcal{M}_n(\mathbb{C}) \cong \mathbb{C}^{n^2}$ on which the center acts trivially, and almost by definition, this gives a faithful representation of $\text{PGL}_n(\mathbb{C})$. (This is the adjoint representation, but you don't need to know anything about Lie algebras or even tangent spaces to recognize that $\text{GL}_n(\mathbb{C})$ embeds into $\mathcal{M}_n(\mathbb{C})$.)