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Nov
emb
er13,
2009                       

Math  221  –  02:  Exam  3                      Name:                                                  

 Prob123456TotalValue121081082270Points       Show  properworkforfull  creditin  problems3-6
 1.(2pts  each)Answertrueorfalse.No  Just
ification  Required.
 (a)              IfAisaninvertible  matrix,  then  thecolumnvectorsofAmust  belinearlyindependent.
   (b)              IfAisanm
×nmatrix,then  row(A)isasubspaceofR

m.    (c)              IfV   isasubspace  ofR
5   

withdimension  3,then  the  orthogonal  complementofV,V
⊥,musthavedimension2.(i.e.  dim(V
⊥)=2)
(d)              Everysubspaceofdimension1inR
2  
isrepresented  byalinethat  goesthroughtheorigin.(e)                     IfAisanm
×nmatrixwithnulli
ty(A)  =0,then  col(A)=R
m   
.   (f)              Ifspan
{~v1,~v
2,…,~v

n }=Rn ,then  theset
{~v1 ,~v2,…,~vn }islinearlyindependent.
     2.(2pt  each)ShortAnswer.  No  Justification  Required.
SupposeAisamatrixwith
       
1    2     3          1    −1   0 row(A)=span
,0        
−5 and  col(A)=span    
1,0
   Givethefollowing:
  (i)
Thesize
ofAis                                 
      
2 1     (ii)
Rank(

A)=                                                (iii
)Nullity(A)=                             (iv)
Rank(AT)=                                             (v)
Nullity(AT)=                         3.(8pts)  LetA~x=~bbealinearsystemofmequations  innunknowns.
Provethat  ifA~x=~bisconsistent,then~bisinthecolumnspaceofA.Proof:

                               4.(10pts)  LetWbeasubspaceofR
n .Showthat  W
⊥  
isasubspaceofRn .Proof: 5.(8pts)  Whatisthedimensionofthesubspace  U=span
{v~1,v~
2,v~3}ofR3,where
v~1=(1,2,1)
v~2=(2,9,0)
v~3=(0,
−5,2) Justify  yourconclusion!
                   −1      1      3
6.GiventhematrixA=
−1   

−5   −3     
0      3      3
−1   −2      0
 (a)  (5pts)  Determine  thereducedrowechelonformofA.
                             (b)  (4pts)  Find  abasisfortherowspaceofA

(c)  (4pts)  Find  abasisforthecolumnspaceofA

               

(d)  (5pts)  Find  abasisforthenullspaceofA

                         

(e)  (4pts)  Verifythatrow(A)
⊥null(A)