Based on a 3 Blue 1 Brown explanation, a matrix of x (ie. $P_c \vec{x}$) , where $P_c = \begin{bmatrix} 2 & -1\\ 1 & 1 \end{bmatrix}$ and $\vec{x} = \begin{bmatrix} 1\\ 2 \end{bmatrix}$, it does two things:
Geometrically: it allows us to transform our grid into someone else's grid(In this example someone = Jennifer).
Numerically: it transforms Jennifer's language into our language. Meaning, exact same vector but, written like how we(in the standard basis) will write them.
My question is, is that logic correct, and can that same logic apply if we sub $\vec{x}$ with $(\vec{x})_c$ like so: $P_c (\vec{x})_c = \vec{x}$?