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

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

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

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