南航双语矩阵论第1章部分习题参考答案.pdf

SolutionKey(chapter1)

Exercise2.

Themainpointistoshowthatthissetisnotclosedundermultiplication.

Take2S,222.But2S.

If2S,thentherearerationalnumbersaandb,suchthat

2a2b3.(Itisclearthata0andb0.)

Thiswillleadto

22

42a3b

6

2ab

Therighthandisarationalnumberandthelefthandsideisanirrationalnumber.This

isimpossible.Thus,Sisnotclosedundermultiplication.Hence,Sisnotafield.

Exercise7.

xyxz

(x)(xy)(x)(xz)

[(x)x]y[(x)x]z

0y0z

yz

Exercise12

Itisavectorspace.

A1：

A2：,

Hence,

A3：Theexistenceofthezeroelement.

Thezeroelementmustsatisfythatforany,

Thatisforany，.Weobtainthatthezeroelementis

A4:Theexistenceofadditiveinverse.

Foreach，itsadditiveinverseis,since

.(Notethatisthezeroelementof)

M1：

M2：

M3：

M4：

Exercise13.

(a)No,itisnotasubspace.DenotethesetbyS.Take

22

p(x)xxSq(x)xxS,.

Then

p(x)q(x)2xS.

Sisnotclosedunderaddition.Hence,Sisnotasubspace.

(Or:ThesetSdoesnotcontainthezeropolynomial,hence,isnotasubspace.)

(b)DenotethesetbyS.

1

(b)Take3,3.

p(x)x1Sp(x)x1S

Thenp(x)q(x)2S.

Sisnotclosedunderaddition.Hence,Sisnotasubspace.

(Or:ThesetSdoesnotcontainthezeropolynomial,hence,isnotasubspace.)

(c)Yes,itisasubspace.Checkthatthissetisclosedunderadditionandscalar

multiplication.

(d)No,itisnotasubspace.Denotethese