MATH 304Linear AlgebraLecture 29:Orthogonal sets.The Gram-Schmidt process.Orthogonal setsLet V be an inner product space with an innerproduct h·, ·i and the induced norm k · k.Definition. A nonempty set S ⊂ V of nonzerovectors is called an orthogonal set if all vectors inS are mutually orthogonal. That is, 0 /∈ S andhx, yi = 0 for any x, y ∈ S, x 6= y.An orthogonal set S ⊂ V is called orthonormal ifkxk = 1 for any x ∈ S.Remark. Vectors v1, v2, . . . , vk∈ V form anorthonormal set if and only ifhvi, vji =1 if i = j0 if i 6= jExamples. • V = Rn, hx, yi = x · y.The standard basis e1= (1, 0, 0, . . . , 0),e2= (0, 1, 0, . . . , 0), . . . , en= (0, 0, 0, . . . , 1).It is an orthonormal set.• V = R3, hx, yi = x · y.v1= (3, 5, 4), v2= (3, −5, 4), v3= (4, 0, −3).v1· v2= 0, v1· v3= 0, v2· v3= 0,v1· v1= 50, v2· v2= 50, v3· v3= 25.Thus the set {v1, v2, v3} is orthogonal but notorthonormal. An orthonormal set is formed bynormalized vectors w1=v1kv1k, w2=v2kv2k,w3=v3kv3k.• V = C [−π, π], hf , gi =Zπ−πf (x)g(x) dx.f1(x) = sin x, f2(x) = sin 2x, . . . , fn(x) = sin nx, . . .hfm, fni =Zπ−πsin(mx) sin(nx) dx=Zπ−π12cos(mx − nx) − cos(mx + nx)dx.Zπ−πcos(kx) dx =sin(kx)kπx=−π= 0 if k ∈ Z, k 6= 0.k = 0 =⇒Zπ−πcos(kx) dx =Zπ−πdx = 2π.hfm, fni =12Zπ−πcos(m − n)x − cos(m + n)xdx=π if m = n0 if m 6= nThus the set {f1, f2, f3, . . . } is orthogonal but notorthonormal.It is orthonormal with respect to a scaled innerproducthhf , g ii =1πZπ−πf (x)g(x) dx.Orthogonality =⇒ linear independenceTheorem Suppose v1, v2, . . . , vkare nonzerovectors that form an orthogonal set. Thenv1, v2, . . . , vkare linearly independent.Proof: Suppose t1v1+ t2v2+ · · · + tkvk= 0for some t1, t2, . . . , tk∈ R.Then for any index 1 ≤ i ≤ k we haveht1v1+ t2v2+ · · · + tkvk, vii = h0, vii = 0.=⇒ t1hv1, vii + t2hv2, vii + · · · + tkhvk, vii = 0By orthogonality, tihvi, vii = 0 =⇒ ti= 0.Orthonormal basesLet v1, v2, . . . , vnbe an orthonormal basis for aninner product space V .Theorem Let x = x1v1+ x2v2+ · · · + xnvnandy = y1v1+ y2v2+ · · · + ynvn, where xi, yj∈ R. Then(i) hx, yi = x1y1+ x2y2+ · · · + xnyn,(ii) kxk =px21+ x22+ · · · + x2n.Proof: (ii) follows from (i) when y = x.hx, yi =*nXi=1xivi,nXj=1yjvj+=nXi=1xi*vi,nXj=1yjvj+=nXi=1nXj=1xiyjhvi, vji =nXi=1xiyi.Let v1, v2, . . . , vnbe a basis for an inner productspace V .Theorem If the basis v1, v2, . . . , vnis anorthogonal set then for any x ∈ Vx =hx, v1ihv1, v1iv1+hx, v2ihv2, v2iv2+ · · · +hx, vnihvn, vnivn.If v1, v2, . . . , vnis an orthonormal set thenx = hx, v1iv1+ hx, v2iv2+ · · · + hx, vnivn.Proof: We have that x = x1v1+ · · · + xnvn.=⇒ hx, vii = hx1v1+ · · · + xnvn, vii, 1 ≤ i ≤ n.=⇒ hx, vii = x1hv1, vii + · · · + xnhvn, vii=⇒ hx, vii = xihvi, vii.Let V be a vector space with an inner product.Suppose that v1, . . . , vk∈ V are nonzero vectorsthat form an orthogonal set. Given x ∈ V , letp =hx, v1ihv1, v1iv1+ · · · +hx, vkihvk, vkivk, o = x − p.Let W denote the span of v1, . . . , vk.Theorem (a) o ⊥ w for all w ∈ W (denoted o ⊥ W ).(b) kok = kx − pk = minw∈Wkx − wk.Thus p is the orthogonal projection of the vectorx on the subspace W . Also, p is closer to x thanany other vector in W , and kok = dist(x, p) is thedistance from x to W .OrthogonalizationLet V be a vector space with an inner product.Suppose x1, x2, . . . , xnis a basis for V . Letv1= x1,v2= x2−hx2, v1ihv1, v1iv1,v3= x3−hx3, v1ihv1, v1iv1−hx3, v2ihv2, v2iv2,. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .vn= xn−hxn, v1ihv1, v1iv1− · · · −hxn, vn−1ihvn−1, vn−1ivn−1.Then v1, v2, . . . , vnis an orthogonal basis for V .The orthogonalization of a basis as described aboveis called the Gram-Schmidt process.NormalizationLet V be a vector space with an inner product.Suppose v1, v2, . . . , vnis an orthogonal basis for V .Let w1=v1kv1k, w2=v2kv2k,. . . , wn=vnkvnk.Then w1, w2, . . . , wnis an orthonormal basis for V .Theorem Any finite-dimensional vector space withan inner product has an orthonormal basis.Remark. An infinite-dimensional vector space withan inner product may or may not have anorthonormal
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