1. ## projective space

Regard the n-dimensional real projective space RPn as the
space of lines in Rn+1 through {0}, i.e.
RPn = (Rn+1 − {0})=∼ with x ∼ y if y = landa x for landa ̸= 0 ∈ R ;
with the equivalence class of x denoted by [x].

(i) Work out the necessary and sufficient condition on a linear map
f : Rn+1 → Rm+1 for the formula [f][x] = [f(x)] to define a map
[f] : RPn → RPm ; [x] → [f(x)] :

(ii) For a linear map f : Rn+1 → Rn+1 satisfying the condition of (i)
prove that the fixed point set
Fix([f]) = {[x] ∈ RPn | [x] = [f(x)] ∈ RPn}
consists of the equivalence classes of the lines in Rn+1 through {0} which contain eigenvectors of f.

(iii) Construct examples of linear maps f : R3 → R3 satisfying the
condition of (i) such that
(a) Fix([f]) is a point.
(b) Fix([f]) is the disjoint union of a point and a circle. [1 Mark]
(c) Fix([f]) is a projective plane.

2. Two things. A) this is hard to read and I doubt most users (myself included) would want to read it. Ask separate questions is separate threads.

Originally Posted by mathshelp06
(b) Fix([f]) is the disjoint union of a point and a circle. [1 Mark]

B) It looks like you forgot to erase

3. Originally Posted by mathshelp06
Regard the n-dimensional real projective space RPn as the
space of lines in Rn+1 through {0}, i.e.
RPn = (Rn+1 − {0})=∼ with x ∼ y if y = landa x for landa ̸= 0 ∈ R ;
with the equivalence class of x denoted by [x].

(i) Work out the necessary and sufficient condition on a linear map
f : Rn+1 → Rm+1 for the formula [f][x] = [f(x)] to define a map
[f] : RPn → RPm ; [x] → [f(x)] :

(ii) For a linear map f : Rn+1 → Rn+1 satisfying the condition of (i)
prove that the fixed point set
Fix([f]) = {[x] ∈ RPn | [x] = [f(x)] ∈ RPn}
consists of the equivalence classes of the lines in Rn+1 through {0} which contain eigenvectors of f.

(iii) Construct examples of linear maps f : R3 → R3 satisfying the
condition of (i) such that
(a) Fix([f]) is a point.
(b) Fix([f]) is the disjoint union of a point and a circle. [1 Mark]
(c) Fix([f]) is a projective plane.
This looks like part of an assessment that counts towards your final grade (my red highlighting). MHF policy is not to knowingly help with such questions. Thread closed.