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November13,2009                        Math  221  -  02:  Exam  3                      Name:                                                  

Show  properworkforfull  creditin  problems3-6

1.(2pts  each)Answertrueorfalse.No  Justification  Required.

(a)              IfAisaninvertible  matrix,  then  thecolumnvectorsofAmust  belinearlyindependent.

(b)              IfAisanm×nmatrix,then  row(A)isasubspaceofRm.

(c)              IfV   isasubspace  ofR5   withdimension  3,then  the  orthogonal  complementofV,V⊥,musthavedimension2.(i.e.  dim(V⊥)=2)

(d)              Everysubspaceofdimension1inR2  isrepresented  byalinethat  goesthroughtheorigin.(e)                     IfAisanm×nmatrixwithnullity(A)  =0,then  col(A)=Rm   .

(f)              Ifspan{~v1,~v2,...,~vn }=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)ThesizeofAis                         

       

       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)  LetWbeasubspaceofRn .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