We can check that PTP= I n by a lengthy computation, or more simply, notice that (P TP) ij = 0 B @ uT 1 u 2 uT 3 1 C A u 1 u 2 u 3 = 0 B @ 1 0 0 0 1 0 0 0 1 1 C A We are using orthonormality of the u i for the matrix multiplication above Orthonormal Change of Basis and Diagonal MatricesAny probability result that is true for unconditional probability remains true if everything is conditioned on some event P ( A ∣ ( B ∩ C)) = P ( A ∩ B ∩ C) P ( B ∩ C) which is what you think the result should be But observe that if you multiply and divide the right side of ( 3) by P ( C)), you can get which is just ( 2)3 talking about this Entertainment Website These Pour Le Doctorat En Medecine Presentee Et Soutenue Le Avril 18 Wellcome Collection Piu control