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Math Help - Is there a list of facts that shows if a matrix is linearly (in)dependent?

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    Is there a list of facts that shows if a matrix is linearly (in)dependent?

    Hi there!

    I was wondering if there is a list of facts showing what makes a linearly independent and linearly dependent matrix?

    For example, if there is a 3x3 matrix with a row of zeros, does that make the matrix linearly dependent?

    For example, how is this matrix linearly independent?

    [16 1]
    [1 17]
    [0 0]
    [0 0]

    Doesn't this have a row of zeros, so it should be dependent?

    the matrix:
    [1 3 -5]
    [5 2 7]
    [0 0 0]

    Is linearly dependent because it has a row of zeros.
    Last edited by Reefer; December 4th 2011 at 09:58 PM.
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: Is there a list of facts that shows if a matrix is linearly (in)dependent?

    Quote Originally Posted by Reefer View Post
    For example, if there is a 3x3 matrix with a row of zeros, does that make the matrix linearly dependent?
    I'm afraid you are confusing concepts. If V is a vector space and v\in V then, \{v\} is linearly independent if and only if v\neq 0 . So, a 3\times 3 matrix is linearly independent if an only if it is different from the zero matrix .

    For example, how is this matrix linearly independent?
    [16 1]
    [1 17]
    [0 0]
    [0 0]
    Doesn't this have a row of zeros, so it should be dependent?
    According to the above comments, it is linearly independent in M_{4\times 2}(\mathbb{R}). Another thing is that its four rows are linearly dependent in \mathbb{R}^2
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    Re: Is there a list of facts that shows if a matrix is linearly (in)dependent?

    Quote Originally Posted by Reefer View Post
    Hi there!

    Doesn't this have a row of zeros, so it should be dependent?

    the matrix:
    [1 3 -5]
    [5 2 7]
    [0 0 0]

    Is linearly dependent because it has a row of zeros.
    Yes because the determinant is zero. Square matrices with a determinant of zero are dependent.
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  4. #4
    MHF Contributor FernandoRevilla's Avatar
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    Re: Is there a list of facts that shows if a matrix is linearly (in)dependent?

    Quote Originally Posted by dwsmith View Post
    Yes because the determinant is zero. Square matrices with a determinant of zero are dependent.
    I suppose you mean that its rows (also its columns) are dependent.
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