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Math Help - Spanning set

  1. #1
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    Spanning set

    This is a really stupid question but for some reason I can't figure it out!!

    Given [1,0,1],[2,1,3] (column vectors). How do I figure out if they are linearly independent or not?
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  2. #2
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    If I recall, in order to be linearly independent, we must be UNABLE to write one as a multiple of the other.

    Let's assume they are linearly DEPENDENT. Thus, there exists some numbee a such that a[1,0,1] = [2,1,3].

    Note that a[1,0,1] = [a,0,a].

    So, we're assuming that [a,0,a] = [2,1,3]. The only way vectors (or matrices) are the same is if each element is equal to the element in the corresponding spot. So we get three equations:

    a = 2
    0 = 1
    a = 3.

    Clearly, the second equation is enough to demonstrate that they aren't linearly dependent. Also, equations 1 and 3, taken together, state that a must be both 2 and 3, also an impossibility.

    So they aren't linearly dependent, which was our assumption. Thus, they MUST be linearly independent.
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  3. #3
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    Quote Originally Posted by pakman View Post
    This is a really stupid question but for some reason I can't figure it out!!

    Given [1,0,1],[2,1,3] (column vectors). How do I figure out if they are linearly independent or not?
    Does there exits a scalar c such that

    \begin{bmatrix} 1 \\ 0 \\ 1 \\ \end{bmatrix} = c \cdot \begin{bmatrix} 2 \\ 1 \\ 3 \\ \end{bmatrix}

    since the answer is no. They are linearly independant
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