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Math Help - Writing a vector as a linear combination.

  1. #1
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    Writing a vector as a linear combination.

    The question is:

    Write (1, 2, 3) belongs to R3 as a linear combination of (1, 1, 1), (1, 0, 1), (1, 0, 0).

    So i wrote:

    (1, 2, 3) = 2(1, 1, 1) + 1(1, 0, 1) - 2(1, 0, 1)

    Which would give:

    2 2 2
    +1 0 1
    = (3, 2, 3)
    - (2, 0, 2)
    = (1, 2, 3)

    So it would appear that it works out, but it seems way to easy. Could someone check this for me and let me know if i am missing something? Thanks.
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  2. #2
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    Quote Originally Posted by GreenDay14 View Post
    The question is:
    Write (1, 2, 3) belongs to R3 as a linear combination of (1, 1, 1), (1, 0, 1), (1, 0, 0).
    So i wrote:
    (1, 2, 3) = 2(1, 1, 1) + 1(1, 0, 1) - 2(1, 0, 1)
    That is the correct solution. It could have been found by solving
    \left( {\begin{array}{*{20}c}<br />
   1 & 1 & 1  \\<br />
   1 & 0 & 0  \\<br />
   1 & 1 & 0  \\<br />
 \end{array} } \right)\left( {\begin{array}{*{20}c}   x  \\   y  \\   z  \\<br />
 \end{array} } \right) = \left( {\begin{array}{*{20}c}<br />
   1  \\   2  \\   3  \\ \end{array} } \right)
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