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Math Help - Linearly Independent Vectors

  1. #1
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    Linearly Independent Vectors

    Again, I think I have come up with the correct answer, but the answer is not provided as theoretical answers are omitted from the answer key (bad system if you ask me.)

    Problem:
    Consider the vectors \vec{v_{1}},\vec{v_{2}},...,\vec{v_{m}} in \mathbb{R}^{n}}, with \vec{v_{m}} = \vec{0}. Are these vectors linearly independent?

    My solution:
    No, they are not. Since \vec{v_{m}} = \vec{0}, rank[\vec{v_{1}} ... \vec{v_{m}}] will fail to equal m. There will not be a pivot in every column, which is a requirement for a set of linearly independent vectors.

    Am I on the right track? Any input appreciated!
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by tangibleLime View Post
    Again, I think I have come up with the correct answer, but the answer is not provided as theoretical answers are omitted from the answer key (bad system if you ask me.)

    Problem:
    Consider the vectors \vec{v_{1}},\vec{v_{2}},...,\vec{v_{m}} in \mathbb{R}^{n}}, with \vec{v_{m}} = \vec{0}. Are these vectors linearly independent?

    My solution:
    No, they are not. Since \vec{v_{m}} = \vec{0}, rank[\vec{v_{1}} ... \vec{v_{m}}] will fail to equal m. There will not be a pivot in every column, which is a requirement for a set of linearly independent vectors.

    Am I on the right track? Any input appreciated!
    I mean, if you want to do it in terms of matrices you're correct, but isn't it just simpler to note that 0v_1+\cdots+v_m=0?
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