# STAT 200 Lecture 14: Untitled4

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Published on 27 Jul 2020
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Solvetheproblem.
1) DeterminewhichofthefollowingsetsisasubspaceofPnforanappropriatevalueofn.
A:Allpolynomialsoftheformp(t)=a+bt2,whereaandbarein
B:Allpolynomialsofdegreeexactly4,withrealcoefficients
C:Allpolynomialsofdegreeatmost4,withpositivecoefficients
A) Conly B) AandBC)Aonly D) Bonly
1)
2) Determinewhichofthefollowingsetsisavectorspace.
Vistheliney=xinthexy-plane:V=x
y
:y=x
y
:y0
Uistheliney=x+1inthexy-plane:U=x
y
:y=x+1
A) Wonly B) UandVC)Vonly D) Uonly
2)
3) LetHbethesetofallpolynomialshavingdegreeatmost4andrationalcoefficients.Determine
whetherHisavectorspace.Ifitisnotavectorspace,determinewhichofthefollowingproperties
itfailstosatisfy.
A:Containszerovector
C:Closedundermultiplicationbyscalars
A) Hisnotavectorspace;notclosedundermultiplicationbyscalars
B) Hisnotavectorspace;doesnotcontainzerovector
C) Hisavectorspace.
3)
4) LetHbethesetofallpolynomialsoftheformp(t)=a+bt2whereaandbareinandb>a.
DeterminewhetherHisavectorspace.Ifitisnotavectorspace,determinewhichofthefollowing
propertiesitfailstosatisfy.
A:Containszerovector
C:Closedundermultiplicationbyscalars
A) Hisnotavectorspace;notclosedundermultiplicationbyscalarsanddoesnotcontainzero
vector
C) Hisnotavectorspace;notclosedundermultiplicationbyscalars
D) Hisnotavectorspace;doesnotcontainzerovector
4)
1
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5) LetHbethesetofallpointsoftheform(s,s-1).DeterminewhetherHisavectorspace.Ifitisnota
vectorspace,determinewhichofthefollowingpropertiesitfailstosatisfy.
A:Containszerovector
C:Closedundermultiplicationbyscalars
A) Hisnotavectorspace;failstosatisfyallthreeproperties
B) Hisnotavectorspace;doesnotcontainzerovector
D) Hisavectorspace.
5)
6) LetHbethesetofallpointsinthexy-planehavingatleastonenonzerocoordinate:
H=x
y
:x,ynotbothzero .DeterminewhetherHisavectorspace.Ifitisnotavectorspace,
determinewhichofthefollowingpropertiesitfailstosatisfy:
A:Containszerovector
C:Closedundermultiplicationbyscalars
B) Hisnotavectorspace;doesnotcontainzerovector
C) Hisnotavectorspace;failstosatisfyallthreeproperties
D) Hisnotavectorspace;doesnotcontainzerovectorandnotclosedundermultiplicationby
scalars
6)
IfthesetWisavectorspace,findasetSofvectorsthatspansit.Otherwise,statethatWisnotavectorspace.
7) Wisthesetofallvectorsoftheform
a+6
b
3b
4a-b
-a
,whereaandbarearbitraryrealnumbers.
A)
1
3
4
-1
,
6
0
-1
0
B)
1
0
4
-1
,
6
3
-1
0
C) Notavectorspace D)
1
0
4
0
,
6
0
-1
0
,
0
3
0
-1
7)
2
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8) Wisthesetofallvectorsoftheform
a-4
b
5
4a+b
-a-b
,whereaandbarearbitraryrealnumbers.
A)
1
0
4
-1
,
-4
5
1
-1
B) Notavectorspace
C)
1
0
4
-1
,
-4
0
1
-1
,
0
5
0
0
D)
1
5
4
-1
,
-4
0
1
-1
8)
Solvetheproblem.
9) Findallvaluesofhsuchthatywillbeinthesubspaceof3spannedbyv1,v2,v3ifv1=
1
2
-4
,
v2=
3
4
-8
,v3=
-1
0
0
,andy=
4
2
h
.
A) h=-4 B) allh-4C)h= -16 D) h=-4or0
9)
DeterminewhetherthevectorubelongstothenullspaceofthematrixA.
10) u=
2
3
1
,A=-23-5
-3-19
A) Yes B) No
10)
11) u=
-1
-2
1
,A=
-2-3-8
-3-1-5
3-20
A) No B) Yes
11)
FindanexplicitdescriptionofthenullspaceofmatrixAbylistingvectorsthatspanthenullspace.
12) A=1-2-2-2
0134
A)
-4
-3
1
0
,
-6
-4
0
1
B)
2
1
0
0
,
2
-3
1
0
,
2
-4
0
1
C)
2
1
0
0
,
-4
-3
1
0
,
-6
-4
0
1
D)
2
-3
1
0
,
2
-4
0
1
12)
3
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