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Math Help - Linear dependence question

  1. #1
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    Linear dependence question

    The question:
    Determine, with reasons whether or not the matricies

    A_1 = \[ \left( \begin{array}{cc}<br />
1 & 2 \\<br />
2 & 1\end{array} \right)\]

    A_2 = \[ \left( \begin{array}{cc}<br />
0 & 1 \\<br />
1 & 0\end{array} \right)\]

    A_3 = \[ \left( \begin{array}{cc}<br />
2 & 1 \\<br />
2 & 1\end{array} \right)\]

    are linearly dependent.

    My attempt:
    Usually with these types of questions I'd create a large matrix and reduce to row echelon form. So I'd start by stating:
    \lambda_1A_1 + \lambda_2A_2 + \lambda_3A_3 = 0

    And create a matrix from there. How do I do this when we have matrices instead of polynomials or vectors? Thanks!
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Glitch View Post
    The question:
    Determine, with reasons whether or not the matricies

    A_1 = \[ \left( \begin{array}{cc}<br />
1 & 2 \\<br />
2 & 1\end{array} \right)\]

    A_2 = \[ \left( \begin{array}{cc}<br />
0 & 1 \\<br />
1 & 0\end{array} \right)\]

    A_3 = \[ \left( \begin{array}{cc}<br />
2 & 1 \\<br />
2 & 1\end{array} \right)\]

    are linearly dependent.

    My attempt:
    Usually with these types of questions I'd create a large matrix and reduce to row echelon form. So I'd start by stating:
    \lambda_1A_1 + \lambda_2A_2 + \lambda_3A_3 = 0

    And create a matrix from there. How do I do this when we have matrices instead of polynomials or vectors? Thanks!
    If it helps (if it doesn't let me know) try considering that two by two matrices are just a fancy representation of four-tuples.
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  3. #3
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    So you're saying to write the matrices as 4-tuples, then form a matrix etc?

    I just tried the following:

    Equating components:
    a_{11} = \lambda_1 + 0 + 2\lambda_3 = 0
    a_{12} = 2\lambda_1 + \lambda_2 + \lambda_3 = 0
    a_{21} = 2\lambda_1 + \lambda_2 + 2\lambda_3 = 0
    a_{22} = \lambda_1 + 0 + \lambda_3 = 0

    Which gives us:

    \[ \left( \begin{array}{ccc}<br />
1 & 0 & 2 \\<br />
2 & 1 & 1 \\<br />
2 & 1 & 2 \\<br />
1 & 0 & 1 \end{array} \right)\]

    Is this correct? Thanks.
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  4. #4
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    Is that approach correct? My final answer was that it was linearly independent. My book doesn't have solutions.
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  5. #5
    MHF Contributor

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    Yes, that is correct.
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