What is the geometric interpretation of a Gram matrix as a linear operator? If I apply a Gram matrix to a vector, what geometric quantity does the resulting vector represent?
A very related question was answered with a geometric interpretation of what a Gram matrix $G$ says about its original, constituent matrix $A$ where $G=A^TA$, but that question does not have any information on what a Gram matrix represents as a linear operator applied to vectors.
The context of my question is in the formula for a projector matrix to a subspace spanned by the linearly independent columns of a matrix $A$, in which the inverse of the Gram matrix of $A$ shows up: $$ P = A(A^TA)^{-1}A^T $$ I understand that the vector quantity $\color{red}{A^T\vec v}$ in the application of $P$ to a vector: $$ P\vec v=A\color{blue}{(A^TA)^{-1}}\color{red}{A^T\vec v} $$ is a vector equal to the dot product of $\vec v$ with each of the columns of $A$, but I can't figure out how to interpret how $\color{blue}{(A^TA)^{-1}}$ then transforms the result of $\color{red}{A^T\vec v}$ (i.e., what that inverse Gram transformation represents).
