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If certain entries of this matrix are all nonzero, show that the only

  1. Feb 25, 2013 #1

    s3a

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    1. The problem statement, all variables and given/known data
    Matrix U is the following matrix:
    http://www.wolframalpha.com/input/?i={{a,b,c},{0,d,e},{0,0,f}}

    Question:
    If a, d, f are all nonzero, show that the only solution to Ux = 0 is x = 0. Then the upper triangular matrix U has independent columns.

    The solution says:
    xcomplete = xparticular + xhomogeneous = (1/2,0,1/2,0) + x2 (-3,1,0,0) + x4(0,0,-2,1)

    2. Relevant equations
    xcomplete = xparticular + xhomogeneous

    3. The attempt at a solution
    I'm very confused; I'm not sure where to start.

    Any help in figuring out what I need to do would be really appreciated!
     
  2. jcsd
  3. Feb 25, 2013 #2

    Dick

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    Re: If certain entries of this matrix are all nonzero, show that the o

    Write out the equations that correspond to Ux=0, putting x=(x1,x2,x3). I think the solution you quoted is for a different problem. U is 3x3 and the solution has four dimensional vectors in it.
     
  4. Feb 25, 2013 #3

    s3a

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    Re: If certain entries of this matrix are all nonzero, show that the o

    Oh, oops.

    Ux = 0

    {{a,b,c},{0,d,e},{0,0,f}} {{x_1},{x_2},{x_3}} = {{0},{0},{0}}

    I can see how this gives z = 0 (since f is the nonzero coefficient of in fz = 0) but, I'm confused for the rest.
     
  5. Feb 25, 2013 #4

    Dick

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    Re: If certain entries of this matrix are all nonzero, show that the o

    Put z=0 into the other equations. What do they become?
     
  6. Feb 25, 2013 #5

    s3a

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    Re: If certain entries of this matrix are all nonzero, show that the o

    Oops. I now see that x = y = z = 0. :)

    So, because x = y = z or x_1 = x_2 = x_3 (depending on the notation chosen) shows that the vector x = 0 and the problem is solved, right?
     
  7. Feb 25, 2013 #6

    Dick

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    Re: If certain entries of this matrix are all nonzero, show that the o

    Yes.
     
  8. Feb 25, 2013 #7

    s3a

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    Re: If certain entries of this matrix are all nonzero, show that the o

    Yay. :)

    Thanks.
     
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