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To represent a standard-coordinate transformation $A$ in a new basis, use the formula $P^{-1}AP$. First convert the input vector to the new basis using $P^{-1}$, apply the transformation $A$ in standard coordinates, then convert the result back to the new basis using $P$.

Conditions: The transformation $A$ is defined in standard coordinates.; The change-of-basis matrix $P$ is invertible.

To represent a standard-coordinate transformation $A$ in a new basis, use the formula $P^{-1}AP$. First convert the input vector to the new basis using $P^{-1}$, apply the transformation $A$ in standard coordinates, then convert the result back to the new basis using $P$.

Conditions: The transformation $A$ is defined in standard coordinates.; The change-of-basis matrix $P$ is invertible.