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Math Help - Linear Algebra

  1. #1
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    Linear Algebra

    My text is not helping at all and the wikipedia explanations of the concepts: nullspace and range are way over my head.

    Given A is a 3x3 matrix
    0 1 0
    0 0 1
    0 0 0

    and B is also a 3x3 matrix
    0 0 0
    1 0 0
    0 1 0

    specify the nullspace of A and B, and the range of A and B

    Thanks
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  2. #2
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    Quote Originally Posted by jblorien View Post
    My text is not helping at all and the wikipedia explanations of the concepts: nullspace and range are way over my head.

    Given A is a 3x3 matrix
    0 1 0
    0 0 1
    0 0 0

    and B is also a 3x3 matrix
    0 0 0
    1 0 0
    0 1 0

    specify the nullspace of A and B, and the range of A and B

    Thanks
    The range is the span of the column vectors

    a \cdot \begin{bmatrix}<br />
0 \\<br />
0 \\<br />
0 \\<br />
\end{bmatrix} + b \cdot \begin{bmatrix}<br />
1 \\<br />
0 \\<br />
0 \\<br />
\end{bmatrix} + c \cdot \begin{bmatrix}<br />
0 \\<br />
1 \\<br />
0 \\<br />
\end{bmatrix}<br /> <br /> <br />

    These vectors span the xy plane so that is your range.

    The null space is the solution to Ax=0

    \begin{bmatrix}<br />
0 && 1 && 0 \\<br />
0 && 0 && 1 \\<br />
0 && 0 && 0 \\<br />
\end{bmatrix} \begin{bmatrix}<br />
x \\<br />
y \\ <br />
z \\<br />
\end{bmatrix} = \begin{bmatrix}<br />
0 \\<br />
0 \\<br />
0 \\<br />
\end{bmatrix}<br /> <br />

    using the augmented matrix we get

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

    So by parameterizing our solution we

    x=t
    y=0
    z=0

    \begin{bmatrix}<br />
t \\<br />
0 \\<br />
0 \\<br />
\end{bmatrix} = t \cdot \begin{bmatrix}<br />
1 \\<br />
0 \\<br />
0 \\<br />
\end{bmatrix}<br /> <br />

    So the null space is all of the points on the x axis.
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  3. #3
    Behold, the power of SARDINES!
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    Note that elementry row operations will change the column space of a matrix.

    the span of the columns is

    a \cdot \begin{bmatrix}<br />
0 \\<br />
1 \\<br />
0 \\<br />
\end{bmatrix}<br />
+b \cdot \begin{bmatrix}<br />
0 \\<br />
0 \\<br />
1 \\<br />
\end{bmatrix}<br />
+c \cdot \begin{bmatrix}<br />
0 \\<br />
0 \\<br />
0 \\<br />
\end{bmatrix}<br /> <br /> <br />

    This is the yz plane.

    Note that elementry row operation will NOT change the null space.

    solving the augmented matrix

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

    in reduced row form

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

    so parameterizing the solution

    x=0
    y=0
    z=t

    so we get...

    \begin{bmatrix}<br />
0 \\<br />
0 \\<br />
t \\<br />
\end{bmatrix} =t \cdot \begin{bmatrix}<br />
0 \\<br />
0 \\<br />
1 \\<br />
\end{bmatrix}<br /> <br />

    So the null space is all points on the z-axis.
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