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7.1E: Inner Products and Norms Exercises

( \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)=x1y1x2y2+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 UV 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,g10f(x)g(x)dx
b. v=f in C[π,π] where f(x)=cosx f,gππf(x)g(x)dx
c. v=[13] in R2 where v,w= vT[1112]w
d. v=[31] in R2,v,w=vT[1112]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=uv
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)=1x;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=ππf(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:VV is an isomorphism of the inner product space V, show that
v,w1=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=[5332]
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]=v1w1v1w2v2w1+
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,2uv
b. u2vw,3wv

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 2u7v,3u+5v.
b. Expand 3u4v,5u+v.
c. Show that u+v2=u2+2u,v+v2.
d. Show that uv2=u22u,v+v2.

Exercise (\PageIndex{23}\) Show that v2+w2= 12{v+w2+vw2} 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+v2=uv2] 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={ww 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 u1, then u,v2v2 for all v in V.
b. (xcosθ+ysinθ)2x2+y2 for all real x,y, and θ.
c. r1v1++rnvn2[r1v1++ rnvn]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=[u2uvuvv2].
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 1v,wvw1, and hence that a unique angle θ exists such that v,wvw=cosθ and 0θπ. 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:
vw=v2+w22vwcosθ.

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.]


7.1E: Inner Products and Norms Exercises is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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