Prof. Dr. Otto May 30, 2011

Linear Algebra II

Exercise Sheet no. 8

Summer term 2011

Prof. Dr. Otto May 30, 2011

Dr. Le Roux Dr. Linshaw

Exercise 1 (Warm-up: orthogonal complement and orthogonal projection)

Let V be a euclidean or unitary vector space of finite dimension, U be a subspace of V and πU : V → U be the orthogonal projection ontoU. Check the following facts.

(a) U^⊥is a subspace ofV.

(b) IfBis a basis ofU, thenU^⊥={v∈V|v⊥B}. (c) πU is linear, surjective andker(πU) =U^⊥. (d) πU◦πU=πU.

(e) For any subspaceW ofV,

π⁻¹_U (W) = (U∩W)⊕U^⊥.

(f) IfB= (v₁, . . . , ,v_n)is an orthonormal basis ofU, then

πU(v) =

n

X

i=1

〈v_i,v〉v_i.

Exercise 2 (Orthogonal complements)

(Exercise 2.3.5 on page 68 of the notes.) LetV be a euclidean or unitary vector space of finite dimension. Moreover, let U,U₁,U₂be subspaces ofV. Prove the following facts.

(a) U₁⊆U₂impliesU₂^⊥⊆U₁^⊥. (b) (U₁+U₂)^⊥=U₁^⊥∩U₂^⊥.

(c) (U^⊥)^⊥=U.

(d) (U₁∩U₂)^⊥=U₁^⊥+U₂^⊥.

Exercise 3 (Stereographic projection)

LetE⊆R³be the plane spanned bye₁ande₂and letS⊆R³be the sphere with radius1and centre0. We denote the north pole ofSbyp:=e₃and we setS_∗:=S\ {p}.

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We define a mapπ:E→S_∗by lettingπ(x)be the point of intersection betweenS_∗and the line passing throughpandx.

(a) Give an explicit formula for π, i.e., find functions f(x,y), g(x,y), and h(x,y) such that π(x,y, 0) = (f(x,y),g(x,y),h(x,y)).

(b) Prove thatπ:E→S_∗is a bijection.

(c) LetC⊆Sbe a circle, i.e., the intersection ofSwith a plane given by an equation of the forma x+b y+cz=d.

Prove that the pre-imageπ⁻¹[C]is either also a circle or a line.

(d) Letc:R→Ebe a line with parametric descriptionxe₁+tv,t∈R, wherev= (cosα, sinα, 0). Note thatcintersects thee₁-axis in the point xe₁under the angleα. Prove that the image ofcunderπ, i.e., the curve π◦c:R→S_∗, intersects the great circle{(u, 0,v)∈S_∗ :u²+^v2=1} under the same angleα. (This implies thatπpreserves angles. Such maps are calledconformal.)

(Hint. Find the angle between the tangent vectors of the two curves. The tangent vector of a curvec at the pointc(t₀)is given by its derivative ^d

dtc _t

0.) Exercise 4 (Characterisations of orthogonal projections)

(Exercise 2.3.2 on page 68 of the notes.) Letϕbe an endomorphism of a finite dimensional euclidean or unitary vector spaceV.

Show the equivalence of the following:

(a) ϕis an orthogonal projection.

(b) ϕ◦ϕ=ϕandker(ϕ)⊥image(ϕ).

(c) ϕ◦ϕ=ϕandv−ϕ(v)⊥ϕ(v)for allv∈V. (d) v−ϕ(v)⊥image(ϕ)for allv∈V.

Exercise 5 (More on orthogonal projections)

(Exercise 2.3.3 on page 68 of the notes.) Show that the orthogonal projections of ann-dimensional euclidean or unitary vector spaceV are precisely those endomorphismsϕofV that are represented w.r.t. a suitable orthonormal basis by a diagonal matrix with ones and zeroes on the diagonal.

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