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$$ \left[ P \right] \left\langle c^\star_P \right\rangle = \left\langle 0,\,  0 ,\,  0,\,  1  \right\rangle^T $$

$$ \left\langle c^\star_P \right\rangle = \left[ P^{-1} \right]  \left\langle 0,\,  0 ,\,  0,\,  1  \right\rangle^T $$

$$ \left[ P \right] \left\langle c_P \right\rangle \ne \left\langle 0,\,  0 ,\,  0,\,  1  \right\rangle^T $$

$$ \left\langle c^\star_P \right\rangle = \left[ T \right] \left\langle c_P \right\rangle $$