Question

if sum of projections is identity then product of any two is zero

Answer 1

E is finite dimensional space and P1,…,Pk are projections i.e. P2i=Pi such thatP1+…+Pk=IdE(∗)I want to prove thatE=k⨁i=1Im(Pi)and then PiPj=0 if i≠j.

Answer 2

E is finite dimensional space and P1,…,Pk are projections i.e. P2i=Pi such thatP1+…+Pk=IdE(∗)I want to prove thatE=k⨁i=1Im(Pi)and then PiPj=0 if i≠j.