What is Projection operator?

Example 5.10 (The combinations of ammonia and water)

In $\ce{NH3}$, $\hat P_{A_1}h_1 \propto h_1 + h_2 + h_3$ (all characters 1). For $E$ (characters $2, -1, 0$), $\hat P_Eh_1 \propto 2h_1 - h_2 - h_3$; projecting $h_2$ and orthogonalising gives the partner $h_2 - h_3$. Normalised: $a_1 = \frac{1}{\sqrt3}(h_1 + h_2 + h_3)$, $e = \frac{1}{\sqrt6}(2h_1 - h_2 - h_3)$, $\frac{1}{\sqrt2}(h_2 - h_3)$. In $\ce{H2O}$, placed in the $xz$ plane, $h_1 + h_2$ is $a_1$ and $h_1 - h_2$ is $b_1$ (it changes sign under $C_2$ and under $\sigma_v'(yz)$, which exchange the hydrogens). With the molecule in the $yz$ plane, as some books prefer, the labels $b_1$ and $b_2$ are exchanged throughout.