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Homework Help: Proof that vectors span R3

  1. Apr 2, 2010 #1

    judahs_lion

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    Show that vectors v1, v2, and v3 span R3.

    V1=(1,0,0)
    V2=(2,2,0)
    V3=(3,3,3)

    I'm pretty sure I'm doing this wrong?

    a(V1) +b(V2) +c(V3) = [x,y,z]

    for (a= 0, b = 0, c = 1/3)

    [0,0,0] +[0,0,0] +[1,1,1] = [x,y,z]

    [1,1,1] = [x,y,z]
     

    Attached Files:

    Last edited: Apr 2, 2010
    judahs_lion, Apr 2, 2010
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  3. Apr 2, 2010 #2

    Mark44

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    Insights Author

    Staff: Mentor

    Not right. In a nutshell you want to show that for an arbitrary vector <x, y, z>, there are some constants a, b, and c so that aV1 +bV2 +cV3 = <x,y,z>.

    You can do this by solving the matrix equation Ab = x for b, where the columns of matrix A are your vectors V1, V2, and V3. The vector I show as b is <a, b, c>, and the vector I show as x is <x, y, z>.
     
    Mark44, Apr 2, 2010
  4. Apr 2, 2010 #3

    VeeEight

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    Try showing that you can generate the standard basis of R3, {(1,0,0), (0,1,0), (0,0,1)}, using the elements v1, v2, v3. For example, what combination of these vectors will give you (0,1,0)?
     
    VeeEight, Apr 2, 2010
  5. Apr 2, 2010 #4

    judahs_lion

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    Is there any way u dumb it down just a lil more , I still feel very lost.
     
    judahs_lion, Apr 2, 2010
  6. Apr 2, 2010 #5

    judahs_lion

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    Ok, I got this far

    SEE ATTACHMENT
     

    Attached Files:

    • dfae.jpg
      dfae.jpg
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    judahs_lion, Apr 2, 2010
  7. Apr 2, 2010 #6

    VeeEight

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    You want to show that {v1, v2, v3} = V spans R3. You already know that the vectors (1,0,0), (0,1,0), and (0,0,1) span R3. So you can try showing that V generates (0,1,0) and (0,0,1) and thus generates R3
     
    VeeEight, Apr 2, 2010
  8. Apr 2, 2010 #7

    judahs_lion

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    How do I know they span R3?
     
    judahs_lion, Apr 2, 2010
  9. Apr 2, 2010 #8

    Mark44

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    Staff: Mentor

    The work in the attachment looks fine. If you can row-reduce your matrix to the identity matrix [1 0 0; 0 1 0; 0 0 1], that's enough to guarantee that your three vectors span R3.
     
    Mark44, Apr 2, 2010
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