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.