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Math Help - Linear Dependent/Spanning System?

  1. #1
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    Linear Dependent/Spanning System?

    Is there a difference between a linear dependent system and a spanning one? I've been confused about this for a while. Something is linearly dependent if it has a non trivial solution, and something is spanning if one of the vectors can be represented as a linear combination of another - implying there is a non-trivial solution- Some of my classmates told me i'm wrong, and I don't understand why. An explanation would be awesome. Can't seem to find anything online.
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  2. #2
    Senior Member jakncoke's Avatar
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    Re: Linear Dependent/Spanning System?

    A Spanning set for a Vector space V can be linearly dependent but not every linearly dependent set is a spanning set for V.
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    Re: Linear Dependent/Spanning System?

    How is that so?
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  4. #4
    Senior Member jakncoke's Avatar
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    Re: Linear Dependent/Spanning System?

    The set of linearly dependent vectors \{ \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix},  \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix},  \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix},  \begin{bmatrix} 3 \\ 0 \\ 0 \end{bmatrix} \} spans  \mathbb{R}^3 . Now the set of linearly dependent vectors \{ \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 3 \\ 0 \\ 0 \end{bmatrix} \} does not span  \mathbb{R}^3
    Last edited by jakncoke; January 24th 2013 at 03:47 PM.
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  5. #5
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    Re: Linear Dependent/Spanning System?

    A system of vectors v1; v2;...; vn in V is called a generating system (also a spanning system, or a complete system) in V if any vector v in V admits representation as a linear combination.

    This is the definition from the book I am using. isn't v2 = 3*v1. Therefore it is a spanning system?
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  6. #6
    Senior Member jakncoke's Avatar
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    Re: Linear Dependent/Spanning System?

    Any vector in V must be admitted as a linear combination.

    So the vector  \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} lives in  \mathbb{R}^3 but give me a linear combination of \{ \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 3 \\ 0 \\ 0 \end{bmatrix} \} to produce r  \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} , cant be done so this linearly dependent system does not span  \mathbb{R}^3
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  7. #7
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    Re: Linear Dependent/Spanning System?

    Ahhh. Thank you very much. I appreciate the help. I guess i have to pay more attention to key words!
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