Question

Projection matrices.

Answer 1

without knowing any thrm that relate to that, it just looks like a mess to me

Answer 2

I got a hint from my teacher:
The targets are orthogonal complements. then use the fact that you have projection matrices.

Answer 3

How about this?

let x be a vector in \( \Re ^7\)
we can decompose it into \( n \in N(A^T) \) and \( c \in C(A) \)
x= n + c

for all x:
PQx = PQ(n+c) = PQn + PQc
Qn = n (n is in the null space of A transpose)
Qc = 0 ( c is not in the null space)
so we have
Pn + 0
Pn = 0 (n is not in the column space of P)
so we have
PQx = 0 for all x --> PQ= O

for all vectors x:
(P+Q)x = (P+Q)(n+c)= Pn + Qn + Pc + Qc
Pn= 0 (n is not in the column space of A)
Qn = n
Pc = c
Qc =0
we get
(P+Q)x = n+c = x

which implies P+Q = I