Say In $\mathbb{R}^3$ I have a point $P_{B_0} = (x, y, z)$ and its basis $B_0 = (\vec{e_0}, \vec{e_1}, \vec{e_2})$, I would like to project this point into a new basis $B_1 = (\vec{f_0}, \vec{f_1}, \vec{f_2})$ as point $P_{B_1} = (x’, y’, z’)$. Please what’s the general formula to get this over with in $\mathbb{R}^n$
I mean a formula that is function of $P_{B_0},B_0, B_1$ in relation to $P_{B_1}$ i.e. something along the lines of $P_{B_1} = \frac{B_1 P_{B_0}}{B_0} $ from $\frac{P_{B_0}}{B_0} = \frac{P_{B_1}}{B_1}$
$ P = B_0 P_{B_0} = B_1 P_{B_1} $
Hence,
$ P_{B_1} = (B_1)^{-1} B_0 P_{B_0} $