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Math Help - Consider 2 homogeneous systems

  1. #1
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    Consider 2 homogeneous systems

    Find all possible solutions to W1 and W2.
    Find the set of all vectors which are solutions to both systems and why it is a vector space.

    Please help.
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  2. #2
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    The augmented matrix for the first is

    \begin{bmatrix}<br />
1 && 2 && 1 && 0 \\<br />
2 && 4 && 2 && 0 \\<br />
1 && 2 && 1 && 0 \\<br />
\end{bmatrix}<br />

    In reduced tow eschelon form you get...

    \begin{bmatrix}<br />
1 && 2 && 1 && 0 \\<br />
0 && 0 && 0 && 0 \\<br />
0 && 0 && 0 && 0 \\<br />
\end{bmatrix}<br />

    so we have two "free" variables we get the solution

    x=-2s-t
    y= s
    z=t

    v_=\begin{bmatrix}<br />
-2s-t \\<br />
s \\<br />
t<br />
\end{bmatrix}= s\begin{bmatrix}<br />
-2 \\<br />
1 \\<br />
0 \\<br />
\end{bmatrix}+ t\begin{bmatrix}<br />
-1 \\<br />
0 \\<br />
1 \\<br />
\end{bmatrix}

    The last two vectors span the solution space of the first matrix.

    Try this proceedure with the second matrix and compare the solution spaces.

    I hope this helps

    Good luck.
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  3. #3
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    thanks man.
    I got the following solution for second one.
    So is S[-2,1,0] the solution for both systems?
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