Consider the linear transformation $T$:$R^4{R}^4$ given by

$T(x,y,z,t)=(y-2z+t,x+2z-t,2y+z,y+t)$

Find linear transformations $P_1,P_2$ and $P_3$ such that

$T={}_1{P}_1+{}_2{P}_2+{}_3{P}_3$ for some ${}_1,{}_2,{}_3$inR,

$P_i^2=P_i$ for $1i3,$

$i{d}_{R^4}=P_1+P_2+P_3,$ and,

$P_i{P}_j=0$ for $1i,j3$ and $ij.$