7.1E: Inner Products and Norms Exercises
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( \newcommand{\kernel}{\mathrm{null}\,}\)
Exercise 7.1E.1 In each case, determine which of axioms P1-P5 fail to hold.
a. V=R2,⟨(x1,y1),(x2,y2)⟩=x1y1x2y2
b.
V=R3,⟨(x1,x2,x3),(y1,y2,y3)⟩=x1y1−x2y2+x3y3
c. V=C,⟨z,w⟩=zˉw, where ˉw is complex conjugation
d. V=P3,⟨p(x),q(x)⟩=p(1)q(1)
e. V=M22,⟨A,B⟩=det(AB)
f. V=F[0,1],⟨f,g⟩=f(1)g(0)+f(0)g(1)
Exercise 7.1E.2 Let V be an inner product space. If U⊆V is a subspace, show that U is an inner product space using the same inner product.
Exercise 7.1E.3 In each case, find a scalar multiple of v that is a unit vector.
a. v=f in C[0,1] where f(x)=x2 ⟨f,g⟩∫10f(x)g(x)dx
b. v=f in C[−π,π] where f(x)=cosx ⟨f,g⟩∫π−πf(x)g(x)dx
c. v=[13] in R2 where ⟨v,w⟩= vT[1112]w
d. v=[3−1] in R2,⟨v,w⟩=vT[1−1−12]w
Exercise 7.1E.4 In each case, find the distance between u and v.
a. u=(3,−1,2,0),v=(1,1,1,3);⟨u,v⟩=u⋅v
b. u=(1,2,−1,2),v=(2,1,−1,3);⟨u,v⟩=u. v
c. u=f,v=g in C[0,1] where f(x)=x2 and g(x)=1−x;⟨f,g⟩=∫10f(x)g(x)dx
d. u=f,v=g in C[−π,π] where f(x)=1 and g(x)=cosx;⟨f,g⟩=∫π−πf(x)g(x)dx
Exercise 7.1E.5 Let a1,a2,…,an be positive numbers. Given v=(v1,v2,…,vn) and w=(w1, w2,…,wn), define ⟨v,w⟩=a1v1w1+…+anvnwn. Show that this is an inner product on Rn.
Exercise 7.1E.6 If {b1,…,bn} is a basis of V and if v=v1b1+⋯+vnbn and w=w1b1+⋯+wnbn are vectors in V, define
⟨v,w⟩=v1w1+⋯+vnwn.
Show that this is an inner product on V.
Exercise 7.1E.7 If p=p(x) and q=q(x) are polynomials in Pn, define
⟨p,q⟩=p(0)q(0)+p(1)q(1)+⋯+p(n)q(n)
Show that this is an inner product on Pn. [Hint for P5: Theorem 6.5.4 or Appendix D.]
Exercise 7.1E.8 Let Dn denote the space of all functions from the set {1,2,3,…,n} to R with pointwise addition and scalar multiplication (see Exercise 35 Section 6.3). Show that ⟨,⟩ is an inner product on Dn if ⟨f,g⟩=f(1)g(1)+f(2)g(2)+ …+f(n)g(n).
Exercise 7.1E.9 Let re (z) denote the real part of the complex number z. Show that ⟨,⟩ is an inner product on C if ⟨z,w⟩=re(zˉw).
Exercise 7.1E.10 If T:V→V is an isomorphism of the inner product space V, show that
⟨v,w⟩1=⟨T(v),T(w)⟩
defines a new inner product ⟨,⟩1 on V.
Exercise (\PageIndex{11}\) Show that every inner product ⟨,⟩ on Rn has the form ⟨x,y⟩=(Ux)⋅(Uy) for some upper triangular matrix U with positive diagonal entries. [Hint: Theorem 8.3.3.]
Exercise (\PageIndex{12}\) In each case, show that ⟨v,w⟩ =vTAw defines an inner product on R2 and hence show that A is positive definite.
a. A=[2111]
b. A=[5−3−32]
c. A=[3223]
d. A=[3446]
Exercise (\PageIndex{13}\) In each case, find a symmetric matrix A such that ⟨v,w⟩=vTAw.
a. ⟨v1v2],[w1w2]⟩=v1w1+2v1w2+
b. ⟨[v1v2],[w1w2]⟩=v1w1−v1w2−v2w1+
c. ⟨[v1v2v3],[w1w2w3]⟩=2v1w1+v2w2+
Exercise (\PageIndex{14}\) If A is symmetric and xTAx= 0 for all columns x in Rn, show that A=0. [Hint: Consider ⟨x+y,x+y⟩ where ⟨x,y⟩=xTAy.]
Exercise (\PageIndex{15}\) Show that the sum of two inner products on V is again an inner product.
Exercise (\PageIndex{16}\) Let ‖u‖=1,‖v‖=2,‖w‖=√3, ⟨u,v⟩=−1,⟨u,w⟩=0 and ⟨v,w⟩=3. Compute:
a. ⟨v+w,2u−v⟩
b. ⟨u−2v−w,3w−v⟩
Exercise (\PageIndex{17}\) Given the data in Exercise 16, show that u+v=w.
Exercise (\PageIndex{18}\) Show that no vectors exist such that ‖u‖=1,‖v‖=2, and ⟨u,v⟩=−3.
Exercise (\PageIndex{19}\) Complete Example 10.1.2.
Exercise (\PageIndex{20}\) Prove Theorem 10.1.1.
Exercise (\PageIndex{21}\) Prove Theorem 10.1.6.
Exercise (\PageIndex{22}\) Let u and v be vectors in an inner product space V.
a. Expand ⟨2u−7v,3u+5v⟩.
b. Expand ⟨3u−4v,5u+v⟩.
c. Show that ‖u+v‖2=‖u‖2+2⟨u,v⟩+‖v‖2.
d. Show that ‖u−v‖2=‖u‖2−2⟨u,v⟩+‖v‖2.
Exercise (\PageIndex{23}\) Show that ‖v‖2+‖w‖2= 12{‖v+w‖2+‖v−w‖2} for any v and w in an inner product space.
Exercise (\PageIndex{24}\) Let ⟨,⟩ be an inner product on a vector space V. Show that the corresponding distance function is translation invariant. That is, show that d(v,w)=d(v+u,w+u) for all v,w, and u in V.
Exercise (\PageIndex{25}\) a. Show that ⟨u,v⟩=14[‖u+v‖2=‖u−v‖2] for all u,v in an inner product space V.
b. If ⟨,⟩and⟨,⟩′ are two inner products on V that have equal associated norm functions, show that ⟨u,v⟩=⟨u,v⟩′ holds for all u and v.
Exercise (\PageIndex{26}\) Let v denote a vector in an inner product space V.
a. Show that W={w∣w in V,⟨v,w⟩=0} is a subspace of V.
b. If V=R3 with the dot product, and if v=(1, −1,2), find a basis for W(W as in (a)).
Exercise (\PageIndex{27}\) Given vectors w1,w2,…,wn and v, assume that ⟨v,wi⟩=0 for each i. Show that ⟨v,w⟩=0 for all w in span{w1,w2,…,wn}.
Exercise (\PageIndex{28}\) If V=span{v1,v2,…,vn} and ⟨v,vi⟩=⟨w,vi⟩ holds for each i. Show that v=w.
Exercise (\PageIndex{29}\) Use the Cauchy-Schwarz inequality in an inner product space to show that:
a. If ‖u‖≤1, then ⟨u,v⟩2≤‖v‖2 for all v in V.
b. (xcosθ+ysinθ)2≤x2+y2 for all real x,y, and θ.
c. ‖r1v1+⋯+rnvn‖2≤[r1‖v1‖+⋯+ rn‖vn‖]2 for all vectors vi, and all ri>0 in R.
Exercise (\PageIndex{30}\) If A is a 2×n matrix, let u and v denote the rows of A.
a. Show that AAT=[‖u‖2u⋅vu⋅v‖v‖2].
b. Show that det(AAT)≥0.
Exercise (\PageIndex{31}\)
a. If v and w are nonzero vectors in an inner product space V, show that −1≤⟨v,w⟩‖v‖‖w‖≤1, and hence that a unique angle θ exists such that ⟨v,w⟩‖v‖‖w‖=cosθ and 0≤θ≤π. This angle θ is called the angle between v and w.
b. Find the angle between v=(1,2,−1,1,3) and w=(2,1,0,2,0) in R5 with the dot product.
c. If θ is the angle between v and w, show that the law of cosines is valid:
‖v−w‖=‖v‖2+‖w‖2−2‖v‖‖w‖cosθ.
Exercise (\PageIndex{32}\) If V=R2, define ‖(x,y)‖=|x|+ |y|.
a. Show that ‖⋅‖ satisfies the conditions in Theorem 10.1.5.
b. Show that ‖⋅‖ does not arise from an inner product on R2 given by a matrix A. [Hint: If it did, use Theorem 10.1.2 to find numbers a, b, and c such that ‖(x,y)‖2=ax2+bxy+cy2 for all x and y.]