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Chapter 5: Q17E (page 261)

Consider a linear space V. For which linear transformations Tfrom Vto is

Rnis<v,w>=T(v).T(w)鈫�Dotproduct

an inner product in V?

Short Answer

Expert verified

T is one-one.

Step by step solution

01

Check the properties.

Here is V is a vector space and we want to find liner transformation T:V鈫�Rnsuch that

v,w=Tv.Twis an inner product.

Let u,v,w鈭�Vand a鈭�R.

Then, consider the following properties.

1)

v,w=Tv.Tw=Tw.Tv=w,v

2)

u+v,w=Tu+v.Tw=Tu+Tv.Tw=Tu.Tw+Tv.Tw=u,w+v,w

3)

av,w=Tav.w=aTv.Tw=av,w

4) Letv鈮�0

v,v=Tv.Tv=Tv2

The above term is not a non-zero.

Hence, the transformation T is one-one.

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Most popular questions from this chapter

If is a symmetric matrix, then must be symmetric as well.

Consider the vector

v炉=[1234]in R4

Find a basis of the subspace of R4consisting of all vectors perpendicular to v鈫�.

a.Consider the matrix product Q1=Q2S, where both Q1and Q2 are n脳mmatrices with orthonormal columns. Show that Sis an orthogonal matrix. Hint: Computelocalid="1659499054761" Q1T=Q1=(Q2S)TQ2S. Note that

Q1T=Q=Q2T=Q2=Im

b.Show that the QRfactorization of an n脳mmatrix Mis unique. Hint: IfQ1R1=Q2R2, thenQ1=Q2R2R1-1 . Now use part (a) and Exercise 50a.

If then脳nmatrices Aand Bare orthogonal, which of the matrices in Exercise 5 through 11 must be orthogonal as well?3A.

Consider a symmetric invertible n脳nmatrix Awhich admits an LDU-factorization A=LDU. See Exercises 90, 93, and 94 of Section 2.4. Recall that this factorization is unique. See Exercise 2.4.94. Show that

U=LT (This is sometimes called theLDLT - factorizationof a symmetric matrix A.)

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