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.