Got a bizzare question, appreciate any hints

[Try hard]

Let V be a finite dimensional vector space, and P <- L(V, V) be a projection, i.e P = P^2

a. Show that I - P is also a projection, that I am P = Ker(I-P) and that
V = the direct sum of I am P and Ker P

b. Suppose that V is also an inner product space; show that
I am P orthognoal to Ker P <=> P = "P transpose conjugate"

c. Show that if P, Q are orthogonal projections, then PQ is a
orthogonal projection <=> PQ = QP, and that in this case
I am PQ = intersection of I am P and I am Q

d. Show that if P, Q are orthogonal projections, then P+Q is an
orthogonal projection <=> PQ = 0, and that in this case
Im(P+Q) = direct sum of I am P and I am Q

 

[matt grime]

Well, what's bizarre?

a is easy computation and applying a standard theorem

these are all 'show from something satisfies a definition'; which bits don't you understand?

b. for instance uses (Px,y)=(x,P^ty) and the definition of orthogonal.

 

[Try hard]

thanks matt, time to review my maths notes

 

[matt grime]

have you met the standard theorem I allude to? it can be shown without appeal to it as well, and you should probably do that. HINT

1 = 1-P+P

where 1 means the identity matrix.

have you done the computation to show 1-P is a projection? (recall a projection is a map Q such that Q^2=Q, or equally, Q(1-Q)=0. what is the characteristic polynomial of Q? what is the minimal poly of Q if Q is neither 1 nor the zero map?)

if you can't see how to start a question it is always advisable to read the notes that tell you the defining properties of what you wish to demonstrate.