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Linear Combinations

  1. Jan 22, 2010 #1
    The problem statement, all variables and given/known data
    Write the vector (1,2,3) as a linear combination of the vectors (1,0,1), (1,0,-1), and (0,1,1).

    The attempt at a solution
    (1,2,3) = C1(1,0,1) + C2(1,0,-1) + C3(0,1,1)

    The matrix for this is:

    [tex]1....1....0....1[/tex]
    [tex]0....0....1....2[/tex]
    [tex]1....-1....1....3[/tex]

    I reduced it to the following:

    [tex]1....0....0....1[/tex]
    [tex]0....1....0....0[/tex]
    [tex]0....0....1....2[/tex]

    Therefore, (1,2,3) is a linear combination of the vectors (1,0,1), (1,0,-1), and (0,1,1). Am I right?
     
    Last edited: Jan 22, 2010
  2. jcsd
  3. Jan 22, 2010 #2

    benorin

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    Easy check:

    [tex]\mbox{Does }\left( \begin{array}{c} 1 \\ 2 \\ 3 \end{array}\right) = \textbf{1}\, \left( \begin{array}{c}1 \\ 0 \\ 1 \end{array}\right) + \, \textbf{0}\, \left( \begin{array}{c}1 \\ 0 \\ -1 \end{array}\right) + \, \textbf{2}\, \left( \begin{array}{c}0 \\ 1 \\ 1 \end{array}\right) \, ?[/tex]​


    You made an arithmedic error reducing the augmented matrix, try again...
     
    Last edited: Jan 22, 2010
  4. Jan 22, 2010 #3

    Char. Limit

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    Do you even need that middle vector...?
     
  5. Jan 22, 2010 #4

    vela

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    Looks right to me.
     
  6. Jan 22, 2010 #5

    vela

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    Yes, it's a linear combination of those vectors, but you should explicitly write out what that linear combination is because that's what the problem asked for.
     
  7. Jan 22, 2010 #6
    Ok, so all I need to do is substitute in C1, C2, and C3 in front of the appropriate vectors in the original equation?
     
  8. Jan 22, 2010 #7

    Mark44

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    Yes. And as benorin mentioned, it's very easy to check.
     
  9. Jan 22, 2010 #8

    benorin

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    One idea conveyed here is that one may use any linearly independent set of vectors to describe a space. Cartesian (sic?) coordinates use the standard basis vectors so that the (x,y,z) style coordinate (1,2,3) is a linear combination of the vectors (1,0,0), (0,1,0), and (0,0,1). Namely,

    (1,2,3) = 1*(1,0,0) + 2*(0,1,0) + 3* (0,0,1)​

    But, other than their linear independence, these are not special. If you have studied linear independence, deter if the 3 vectors used in the problem match this requirement. They needn't even be boring, stick-arrow vectors, polar, cylindrical, spherical coordinates also work.
     
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