change-of-coordinates matrix

The matrix $P$ converting $\mathcal B$ -coordinates to standard coordinates.

For a basis $\mathcal B=\{\mathbf b_1,\dots,\mathbf b_n\}$ of $\mathbb R^n$ , the change-of-coordinates matrix from $\mathcal B$ to the standard basis, $P$ , has columns $\mathbf b_1,\dots,\mathbf b_n$ in standard coordinates, so $\mathbf x = P[\mathbf x]_{\mathcal B}$ . Because $\mathcal B$ 's vectors are independent, $P$ is invertible, and $[\mathbf x]_{\mathcal B} = P^{-1}\mathbf x$ runs the conversion the other way.