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Math Help - Nullspace

  1. #1
    Rui
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    Nullspace

    Ax=b
    a) calculate the nullspace Null(A) and
    b) give the general solution of the system Ax=b or state that the system inconsistent.
    For b) write in the form x= p+y where p is a particular solution and y ranges across Null (A)

    A=1 1 0 2
    2 1 1 -2
    2 2 2 1

    b= 12
    0
    14
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  2. #2
    Behold, the power of SARDINES!
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    Quote Originally Posted by Rui View Post
    Ax=b
    a) calculate the nullspace Null(A) and
    b) give the general solution of the system Ax=b or state that the system inconsistent.
    For b) write in the form x= p+y where p is a particular solution and y ranges across Null (A)

    A=1 1 0 2
    2 1 1 -2
    2 2 2 1

    b= 12
    0
    14
    To find the Null space solve the matrix equation Ax=0

    \begin{bmatrix}<br />
1 && 1 && 0 && 2 && 0 \\<br />
2 && 1 && 1 && -2 && 0\\<br />
2 && 2 && 2 && 1 && 0 \\<br />
\end{bmatrix} =  \begin{bmatrix}<br />
1 && 0 && 0 && -5/2 && 0 \\<br />
0 && 1 && 0 && 9/2 && 0\\<br />
0 && 0 && 1 && -3/2 && 0 \\<br />
\end{bmatrix}

    now is x_4=t then x_3=3t/2
    x_2=-9t/2  x_1=5t/2

    so the vector that spans the Null Space is

    v= \begin{bmatrix}<br />
5t/2 \\<br />
-9t/2 \\<br />
3t/2 \\<br />
t \\<br />
\end{bmatrix} = t/2 \begin{bmatrix}<br />
5 \\<br />
-9 \\<br />
3 \\<br />
2 \\<br /> <br />
\end{bmatrix} = s \begin{bmatrix}<br />
5 \\<br />
-9 \\<br />
3 \\<br />
2 \\<br /> <br />
\end{bmatrix}<br /> <br />

    solving the system Ax=b

    \begin{bmatrix}<br />
1 && 1 && 0 && 12 && 0 \\<br />
2 && 1 && 1 && 0 && 0\\<br />
2 && 2 && 2 && 14 && 0 \\<br />
\end{bmatrix}  =  \begin{bmatrix}<br />
1 && 0 && 0 && -5/2 && -7 \\<br />
0 && 1 && 0 && 9/2 && 19\\<br />
0 && 0 && 1 && -3/2 && -5 \\<br />
\end{bmatrix}

    A particualr solution is
    \begin{bmatrix}<br />
-7 \\<br />
19 \\<br />
-5 \\<br />
0 \\<br />
\end{bmatrix}

    so finally we get all solutions are of the form

    \begin{bmatrix}<br />
x_1 \\<br />
x_2 \\<br />
x_3 \\<br />
x_4 \\<br />
\end{bmatrix}= s \begin{bmatrix}<br />
5 \\<br />
-9 \\<br />
3 \\<br />
2 \\<br /> <br />
\end{bmatrix}+ \begin{bmatrix}<br />
-7 \\<br />
19 \\<br />
-5 \\<br />
0 \\<br />
\end{bmatrix}<br />
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