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Math Help - finding the basis of a homogeneous system

  1. #1
    Member Jskid's Avatar
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    Jul 2010
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    finding the basis of a homogeneous system

    Find a basis for and the dimensions of the solution space of the given homogeneous system.
    x_1-x_2+2x_3+3x_4+4x_5=0
    -x_1+2x_2+3x_3+4x_4+5x_5=0
    x_1-x_2+3x_3+5x_4+6x_5=0
    3x_1-4x_2+1x_3+2x_4+3x_5=0

    So I make a matrix with these coefficients and rref to give
    <br />
\[<br />
\left[ {\begin{array}{ccccc}<br />
 1 & 0 & 0 & 0 & \frac{1}{3}  \\<br />
 0 & 1 & 0 & 0 & 0  \\<br />
 0 & 0 & 1 & 0 & \frac{4}{3} \\<br />
 0 & 0 & 0 & 1 & \frac{1}{3} \\<br />
 \end{array} } \right]<br />
\]

    (Here's the part I'm not sure if I did right)
    Every solution is of the form <br />
\[<br />
\left[ {\begin{array}{c}<br />
 \frac{-1}{3}r  \\<br />
 0  \\<br />
 \frac{-4}{3}r \\<br />
 \frac{-1}{3}r \\<br />
 \end{array} } \right]<br />
\] where r is any real number. The dimension is 1.
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  2. #2
    Banned
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    Oct 2009
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    Quote Originally Posted by Jskid View Post
    Find a basis for and the dimensions of the solution space of the given homogeneous system.
    x_1-x_2+2x_3+3x_4+4x_5=0
    -x_1+2x_2+3x_3+4x_4+5x_5=0
    x_1-x_2+3x_3+5x_4+6x_5=0
    3x_1-4x_2+1x_3+2x_4+3x_5=0

    So I make a matrix with these coefficients and rref to give
    <br />
\[<br />
\left[ {\begin{array}{ccccc}<br />
 1 & 0 & 0 & 0 & \frac{1}{3}  \\<br />
 0 & 1 & 0 & 0 & 0  \\<br />
 0 & 0 & 1 & 0 & \frac{4}{3} \\<br />
 0 & 0 & 0 & 1 & \frac{1}{3} \\<br />
 \end{array} } \right]<br />
\]

    (Here's the part I'm not sure if I did right)
    Every solution is of the form <br />
\[<br />
\left[ {\begin{array}{c}<br />
 \frac{-1}{3}r  \\<br />
 0  \\<br />
 \frac{-4}{3}r \\<br />
 \frac{-1}{3}r \\<br />
 \end{array} } \right]<br />
\] where r is any real number. The dimension is 1.


    It's correct, assuming your rref is correct.

    Tonio
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