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Use the inverse of the change-of-basis matrix $P$. If $P$ converts from alternative to standard coordinates, then $P^{-1}$ converts from standard to alternative coordinates.

Conditions: The change-of-basis matrix $P$ is invertible.; The input vector is given in standard coordinates.

Use the inverse of the change-of-basis matrix $P$. If $P$ converts from alternative to standard coordinates, then $P^{-1}$ converts from standard to alternative coordinates.

Conditions: The change-of-basis matrix $P$ is invertible.; The input vector is given in standard coordinates.

Construct a matrix $P$ whose columns are the alternative basis vectors expressed in standard coordinates. Multiplying $P$ by a coordinate vector in the alternative basis yields the standard coordinates of that vector.

Conditions: The alternative basis vectors are known in standard coordinates.; The input vector is given as coordinates in the alternative basis.

Construct a matrix $P$ whose columns are the alternative basis vectors expressed in standard coordinates. Multiplying $P$ by a coordinate vector in the alternative basis yields the standard coordinates of that vector.

Conditions: The alternative basis vectors are known in standard coordinates.; The input vector is given as coordinates in the alternative basis.

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.