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Linear combinations?

  1. May 6, 2012 #1
    Hi


    If i have 3 4x1 matrices and i want to check if i can express a 4th matrix as the linear combination of the first 3.


    my 3 vectors:

    1 7 -2
    4 10 1
    2 -4 5
    -3 -1 -4

    can this vector be expressed a linear combination of the first 3:

    54
    0
    -108
    78


    my attempt:

    i made a big matrix out of them:

    1 7 -2 c1 54
    4 10 1 c2 0
    2 -4 5 c3 -108
    -3 -1 -4 c4 78



    i do gaussian elimination:

    1 0 1.5 -30
    0 1 -.5 12

    or

    c1 + 1.5c2 = -30
    c2 - .5c3


    not sure what to do now.
     
  2. jcsd
  3. May 7, 2012 #2

    micromass

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    So your vector is a linear combination if and only if there exists [itex]\alpha,\beta,\gamma\in \mathbb{R}[/itex] such that

    [tex]\left\{
    \begin{array}{l}
    \alpha + 7\beta -2\gamma = 54\\
    4\alpha+ 10\beta +\gamma = 0\\
    2\alpha -4\beta +5\gamma= -108\\
    -3\alpha-\beta -4\gamma = 78
    \end{array}
    \right.[/tex]

    Due to Gaussian elimination (which I did not check) you reduced this question. That is: the vector is a linear combination if and only if there exists [itex]\alpha,\beta,\gamma\in\mathbb{R}[/itex] such that

    [tex]\left\{
    \begin{array}{l}
    \alpha +1.5\gamma= -30\\
    \beta -.5\gamma = 12
    \end{array}
    \right.[/tex]

    Can you find a suitable [itex]\alpha,\beta,\gamma[/itex] now?? Just put [itex]\gamma=1[/itex] and see what the [itex]\alpha[/itex] and [itex]\beta[/itex] are.
     
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