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^{n}as the

space of lines in

^{Rn+1}through {0}, i.e.

RP

^{n}= (R

^{n+1}− {0}) /~ with x ~ y if y = λx for λ not equal to 0 ∈ R ;with the equivalence class of x denoted by [x].

(i) Work out the necessary and sufficient condition on a linear map

f : R

^{n+1}→ R

^{ m+1}for the formula [f][x] = [f(x)] to define a map

[f]: RP

^{n}→ RP

^{m}; [x] → [f(x)] :

(ii) For a linear map f : R

^{n+1}→ R

^{n+1}satisfying the condition of (i)

prove that the fixed point set

Fix([f]) = {[x] ∈ RP

^{n}| [x] = [f(x)] ∈ RP

^{n}}

consists of the equivalence classes of the lines in R

^{n+1 }through {0} which contain eigenvectors of f.

(iii) Construct examples of linear maps f : R

^{3}→ R

^{3 }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.

(c) Fix([f]) is a projective plane.