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Math Help - Linear independence problem

  1. #1
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    Linear independence problem

    If A\in\mathbb{R}^{4X8} then any 6 columns of A are linearly independent.

    True or False?

    thankyou
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  2. #2
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    Quote Originally Posted by Bucephalus View Post
    If A\in\mathbb{R}^{4X8} then any 6 columns of A are linearly independent.

    True or False?

    thankyou
    Pick the zero matrix.
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  3. #3
    Senior Member Twig's Avatar
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    Also, in general.

     A\in\mathbb{R}^{mXn}

    The column vectors of A are in  \mathbb{R}^{m} .
    If there are more vectors than there are entries in each vector, the vectors cannot be linearly independent.
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  4. #4
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    Pick the zero matrix?
    What does that mean?
    Can I have a little more information please?
    Are you saying that if the matrix is all zeros, then each column is a linear combination of another?
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  5. #5
    Senior Member Twig's Avatar
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    He is saying that if you have  A\in \mathbb{R}^{4 \times 8} , then you have an arbitrary matrix, with column vectors in  \mathbb{R}^{4} .

    ANY set of vectors containing the zero vector is linearly dependent, because:
     S = {u_{1} ... u_{p}} - A set of vectors, and the zero vector being in S, then:

     \vec{0} = c_{1}\cdot \vec{u_{1}} + ... + c_{j} \cdot \vec{0} + ... + c_{p} \cdot \vec{p}
    With all  c_{i} = 0 \mbox { except } c_{j}
    Hence there is a nontrivial solution for  A\vec{x} = \vec{b} .

    Now, is the zero vector in the zero matrix ?
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  6. #6
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    Thanks for the explanation.
    Yes it is.
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