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Math Help - Linear Algebra understanding question

  1. #1
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    Linear Algebra understanding question

    find the vector \vec{v}\in \mathbb{R}^3
    who meets the requirements:

    for every matrix A=\begin{bmatrix}<br />
a_1_1 & a_1_2 & a_1_3\\ <br />
a_2_1 & a_2_2 & a_2_3\\ <br />
a_3_1 & a_3_2 & a_3_3<br />
\end{bmatrix}

    exist:
    A\vec{v}=\begin{bmatrix}<br />
a_1_2-a_1_3 \\ <br />
a_2_2-a_2_3 \\ <br />
a_3_2-a_3_3 <br />
\end{bmatrix}

    Can some one please help me here? is \vec{v} is eigenvector?!
    Thank you!
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  2. #2
    Senior Member
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    Dec 2010
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    Define your vector to be
    <br />
\vec{v}=\begin{bmatrix}<br />
x \\<br />
y \\<br />
z<br />
\end{bmatrix}<br />

    And look at the value of A\vec{v} algebraically. You should be able to get exactly what \vec{v} should be.

    For example, the first row of A\vec{v} should be a_{11}x + a_{12}y + a_{13}z, and this is equal to a_{12} - a_{13}.
    What are  x, y, z?
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  3. #3
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    it seems to me that is should be \begin{bmatrix}<br />
0\\ <br />
1\\ <br />
-1<br />
\end{bmatrix}
    am i right?

    and i cold just write it like that?
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  4. #4
    Senior Member
    Joined
    Dec 2010
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    Yes, that is the only possible vector for the solution.

    If you are unsure pick values for A that make A invertible (easiest would be the identity matrix).

    Now substituting values, we have:

    <br />
\begin{bmatrix}<br />
1 & 0 & 0\\<br />
0 & 1 & 0\\<br />
0 & 0 & 1<br />
\end{bmatrix}<br />
\vec{v}=\begin{bmatrix}<br />
0 - 0 \\<br />
1 - 0 \\<br />
0 - 1<br />
\end{bmatrix}<br />
    Which gives you the unique solution.
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  5. #5
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    Oct 2010
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    Ok i got it!
    Thank you very much!
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