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Math Help - Understanding linear combinations and independence

  1. #1
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    Question Understanding linear combinations and independence

    Guys, I've looked in a few places to try and get an understanding of this, and I'm pretty confused to be perfectly honest.

    From what I can understand for vectors to be linear dependent, they have to be able to be put in a linear combination.

    I'm having trouble how to work out how to put vectors into a linear combination.
    For instance on wikipedia, http://en.wikipedia.org/wiki/Linearly

    <br />
\left( {\begin{array}{*{20}c}<br />
   0 & 0 & 1 & 4  \\<br />
   0 & 2 & { - 2} & 2  \\<br />
   1 & { - 2} & 1 & 3  \\<br />
\end{array}} \right)<br />

    They show 3 vectors as being independent, but the total of 4 vectors being dependent, I can understand the 4 vector case as in R^3 can only have 3 vectors being independent of each other. However I don't understand how the determine that the first 3 vectors are independent

    They show a linear combination of
    v_1  = \left( {\frac{{ - 5}}{9}} \right)v_2  + \left( {\frac{{ - 4}}{9}} \right)v_3  + \left( {\frac{1}{9}} \right)v_4

    They give a description of why this is a linear combination but I don't seem to follow that either. Would be very grateful for a explanation of the going on here. Thanks
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  2. #2
    Super Member Showcase_22's Avatar
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    They're using simultaneous equations.

    v_1=\alpha_2v_2+\alpha_3v_3+\alpha_4v_4 and solving for each of the \alpha_i's.
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