Definition 12.20 LetVbe a Euclidean or unitary vector space with the scalar product
·,·, and letU ⊆Vbe a subspace. Then
U⊥:= {v∈V| v,u =0 for allu ∈U} is called theorthogonal complement ofU(inV).
Lemma 12.21 The orthogonal complementU⊥is a subspace ofV.
Proof Exercise. ⊓⊔
Lemma 12.22 IfV is an n-dimensional Euclidean or unitary vector space, and if U ⊆Vis an m-dimensional subspace, thendim(U⊥)=n−m andV =U⊕U⊥. Proof We know thatm≤n(cp. Lemma9.27). Ifm=n, thenU =V, and thus
U⊥=V⊥= {v∈V| v,u =0 for allu∈V} = {0}, so that the assertion is trivial.
Thus letm <n and let{u1, . . . ,um}be an orthonormal basis ofU. We extend this basis to a basis ofVand apply the Gram-Schmidt method in order to obtain an orthonormal basis{u1, . . . ,um,um+1, . . . ,un}ofV. Then span{um+1, . . . ,un} ⊆U⊥ and thereforeV = U+U⊥. Ifw ∈ U∩U⊥, thenw, w =0, and hencew =0, since the scalar product is positive definite. Thus,U∩U⊥= {0}, which implies that V = U⊕U⊥ and dim(U⊥) = n−m(cp. Theorem9.29). In particular, we have
U⊥=span{um+1, . . . ,un}. ⊓⊔
12.3 The Vector Product inR3,1 183
we can write the vector product as v×w=det
ν2 ω2
ν3 ω3
e1−det
ν1ω1
ν3ω3
e2+det
ν1ω1
ν2ω2
e3.
Lemma 12.24 The vector product is linear in both components and for allv, w∈ R3,1the following properties hold:
(1) v×w= −w×v, i.e., the vector product isanti commutativeoralternating.
(2) v×w 2 = v 2 w 2− v, w2, where·,·is the standard scalar product and · the Euclidean norm ofR3,1.
(3) v, v×w = w, v×w =0, where·,·is the standard scalar product ofR3,1.
Proof Exercise. ⊓⊔
By (2) and the Cauchy-Schwarz inequality (12.2), it follows thatv×w=0 holds if and only ifv, ware linearly dependent. From (3) we obtain
λv+µw, v×w =λv, v×w +µw, v×w =0,
for arbitraryλ,µ ∈R. Ifv, ware linearly independent, then the productv×wis orthogonal to the plane through the origin spanned byvandwinR3,1, i.e.,
v×w ∈ {λv+µw|λ,µ∈R}⊥. Geometrically, there are two possibilities:
The positions of the three vectorsv, w, v×won the left side of this figure correspond to the “right-handed orientation” of the usual coordinate system ofR3,1, where the canonical basis vectorse1,e2,e3are associated with thumb, index finger and middle finger of the right hand. This motivates the nameright-hand rule. In order to explain this in detail, one needs to introduce the concept oforientation, which we omit here.
Ifϕ∈ [0,π]is the angle between the vectorsv, w, then v, w = v w cos(ϕ)
(cp. Definition12.7) and we can write (2) in Lemma12.24as
v×w 2= v 2 w 2− v 2 w 2 cos2(ϕ)= v 2 w 2 sin2(ϕ), so that
v×w = v w sin(ϕ).
A geometric interpretation of this equation is the following:The norm of the vector product ofv andwis equal to the area of the parallelogram spanned byvandw.
This interpretation is illustrated in the following figure:
Exercises
12.1 LetV be a finite dimensional real or complex vector space. Show that there exists a scalar product onV.
12.2 Show that the maps defined in Example12.2are scalar products on the cor- responding vector spaces.
12.3 Let·,·be an arbitrary scalar product onRn,1. Show that there exists a matrix A∈Rn,nwithv, w =wTAvfor allv, w∈Rn,1.
12.4 LetV be a finite dimensionalR- orC-vector space. Lets1ands2 be scalar products onVwith the following property: Ifv, w∈Vsatisfys1(v, w)=0, then alsos2(v, w) =0. Prove or disprove: There exists a real scalarλ>0 withs1(v, w)=λs2(v, w)for allv, w∈V.
12.5 Show that the maps defined in Example12.4are norms on the corresponding vector spaces.
12.6 Show that
A 1= max
1≤j≤m
n i=1
|ai j| and A ∞= max
1≤i≤n
m j=1
|ai j|
for allA= [ai j] ∈Kn,m, whereK =RorK =C(cp. (6) in Example12.4).
12.7 Sketch for the matrixAfrom (6) in Example12.4andp∈ {1,2,∞}, the sets {Av|v∈R2,1, v p =1} ⊂R2,1.
12.8 LetVbe a Euclidean or unitary vector space and let · be the norm induced by a scalar product onV. Show that · satisfies theparallelogram identity
v+w 2+ v−w 2 =2( v 2+ w 2) for allv, w∈V.
12.3 The Vector Product inR3,1 185
12.9 LetVbe aK-vector space (K =RorK =C) with the scalar product·,· and the induced norm · . Show thatv, w∈V are orthogonal with respect to·,·if and only if v+λw = v−λw for allλ∈K.
12.10 Does there exist a scalar product·,·onCn,1, such that the 1-norm ofCn,1 (cp. (5) in Example12.4) is the induced norm by this scalar product?
12.11 Show that the inequality n
i=1
αiβi
2
≤ n
i=1
(γiαi)2 · n
i=1
βi
γi
2
holds for arbitrary real numbersα1, . . . ,αn,β1, . . . ,βnand positive real num- bersγ1, . . . ,γn.
12.12 LetVbe a finite dimensional Euclidean or unitary vector space with the scalar product·,·. Let f :V →V be a map withf(v), f(w) = v, wfor all v, w∈V. Show that f is an isomorphism.
12.13 Let V be a unitary vector space and suppose that f ∈ L(V,V) satisfies f(v), v =0 for allv∈V. Prove or disprove that f =0.
Does the same statement also hold for Euclidean vector spaces?
12.14 LetD =diag(d1, . . . ,dn)∈ Rn,nwithd1, . . . ,dn >0. Show thatv, w = wTDvis a scalar product onRn,1. Analyze which properties of a scalar product are violated if at least one of thediis zero, or when alldiare nonzero but have different signs.
12.15 Orthonormalize the following basis of the vector spaceC2,2with respect to the scalar productA,B =trace(BHA):
&
1 0 0 0
, 1 0
0 1
, 1 1
0 1
, 1 1
1 1 '
.
12.16 LetQ∈Rn,nbe an orthogonal or letQ∈Cn,nbe a unitary matrix. What are the possible values of det(Q)?
12.17 Letu ∈Rn,1\ {0}and let
H(u)=In−2 1
uTuuuT ∈ Rn,n.
Show that then columns of H(u)form an orthonormal basis ofRn,1 with respect to the standard scalar product. (Matrices of this form are calledHouse- holder matrices.9We will study them in more detail in Example18.15.) 12.18 Prove Lemma12.21.
9Alston Scott Householder (1904–1993), pioneer of Numerical Linear Algebra.
12.19 Let
[v1, v2, v3] =
⎡
⎢⎣
√1 2 0 √1
2
−√12 0 √1 2
0 0 0
⎤
⎥⎦∈R3,3.
Analyze whether the vectorsv1, v2, v3are orthonormal with respect to the stan- dard scalar product and compute the orthogonal complement of span{v1, v2, v3}. 12.20 LetVbe a Euclidean or unitary vector space with the scalar product·,·, let
u1, . . . ,uk∈V and letU =span{u1, . . . ,uk}. Show that forv∈V we have v∈U⊥if and only ifv,uj =0 for j =1, . . . ,k.
12.21 In the unitary vector spaceC4,1 with the standard scalar product let v1 = [−1,i,0,1]T andv2 = [i,0,2,0]T be given. Determine an orthonormal basis of span{v1, v2}⊥.
12.22 Prove Lemma12.24.
Chapter 13
Adjoints of Linear Maps
In this chapter we introduce adjoints of linear maps. In some sense these represent generalizations of the (Hermitian) transposes of a matrices. A matrix is symmetric (or Hermitian) if it is equal to its (Hermitian) transpose. In an analogous way, an endomorphism is selfadjoint if it is equal to its adjoint endomorphism. The sets of symmetric (or Hermitian) matrices and of selfadjoint endomorphisms form certain vector spaces which will play a key role in our proof of the Fundamental Theorem of Algebra in Chap.15. Special properties of selfadjoint endomorphisms will be studied in Chap.18.