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Chapter 5: Q19E (page 263)

There exists a subspace Vof鈩�5 such thatdim(V)=dim(V鈯�) whereV鈯� denotes the orthogonal complementof V.

Short Answer

Expert verified

False

Step by step solution

01

Step by step solution Step 1: Orthogonal complement

In themathematicalfields oflinear algebraandfunctional analysis, the orthogonal complement of asubspaceW of avector spaceV equipped with abilinear formB is the setQTQ=QQT=Iof all vectors in V that areorthogonalto every vector in W. Informally, it is called the prep, short for perpendicular complement. It is a subspace of V.

02

Determine whether the statement is true or false

For example,

Let V be a vector space over a field F equipped with a bilinear form B. We define u to be left-orthogonal to V, and v to be right-orthogonal to u when B(u, v)=0.

For a subset W of Vdefine the left orthogonal complement W鈯�to be:

W鈯�={x鈭�V:B(x,y)=0

Thus,

dim(V)+dim(V鈯�)=5

Which means that one is even and the other one is odd.

Hence, the statement is false.

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Most popular questions from this chapter

Leonardo da Vinci and the resolution of forces. Leonardo (1452鈥�1519) asked himself how the weight of a body, supported by two strings of different length, is apportioned between the two strings.

Three forces are acting at the point D: the tensions F1and F2in the strings and the weight W鈫�. Leonardo believed that

||F鈫�1||||F鈫�2||=E炉A炉E炉B炉

Was he right? (Source: Les Manuscripts de L茅onard de Vinci, published by Ravaisson-Mollien, Paris, 1890.)

Hint: ResolveF1into a horizontal and a vertical component; do the same for F2. Since the system is at rest, the equationF鈫�1+F鈫�2+W鈫�=0鈫�holds. Express the ratios

||F鈫�1||||F鈫�2||and E炉A炉E炉B炉. In terms of伪and 尾, using trigonometric functions, and compare the results.

Question: If the n脳nmatrices Aand Bare symmetric and Bis invertible, which of the matrices in Exercise 13 through 20 must be symmetric as well?A10.

Consider a basisv鈫�1,v鈫�2,...v鈫�m of a subspaceVofrole="math" localid="1659434380505" 鈩�n. Show that a vectorx鈫� inrole="math" localid="1659434402220" 鈩�nis orthogonal toV if and only ifx鈫� is orthogonal to all vectorsv鈫�1,v鈫�2,...v鈫�m.

If A and B are arbitraryn脳n matrices, which of the matrices in Exercise 21 through 26 must be symmetric?

A-AT.

Using paper and pencil, find the QR factorization of the matrices in Exercises 15 through 28. Compare with Exercises 1 through 14.

20.[235046007]
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