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Math Help - [SOLVED] Basis for the subpace V= Span S of R^3?Any ideas?

  1. #1
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    [SOLVED] Basis for the subpace V= Span S of R^3?Any ideas?

    This is a two part question:

    a) Let S= (v1,v2,v3) , where:
    v1 = (1,2,3) v2 = (2,1,4) v3 = (1,1,2)
    Find a basis for the subspace V= span S of R^3.

    I thought I understood how to do this but now I think I'm weong because the second part doesn't apply to my answer. I expressed the vectors as the columns of a matrix and found the basis of the column space to be all 3 vectors which are then the basis for span S. But the second part states:

    b) Express each vector not in the basis as a linear combination of the found basis vector.

    With my answer all 3 vectors are in the basis so where did I go wrong?

    Any help would be appreciated.

    Thanks in advance.
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  2. #2
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    Quote Originally Posted by chocaholic View Post
    This is a two part question:

    a) Let S= (v1,v2,v3) , where:
    v1 = (1,2,3) v2 = (2,1,4) v3 = (1,1,2)
    Find a basis for the subspace V= span S of R^3.

    I thought I understood how to do this but now I think I'm weong because the second part doesn't apply to my answer. I expressed the vectors as the columns of a matrix and found the basis of the column space to be all 3 vectors which are then the basis for span S. But the second part states:

    b) Express each vector not in the basis as a linear combination of the found basis vector.

    With my answer all 3 vectors are in the basis so where did I go wrong?

    Any help would be appreciated.

    Thanks in advance.

    \left|\begin{pmatrix}1&2&3\\2&1&4\\1&1&2\end{pmatr  ix}\right| =2+8+6-4-8-3=1\neq 0\Longrightarrow the set of vectors \{(1,2,3)\,,\,(2,1,4)\,,\,(1,1,2)\} is linearly independent and is thus a basis for \mathbb{R}^3 ,which

    of course means that in fact S=\mathbb{R}^3 and thus you went wrong nowhere.

    I suppose part (b)'s answer could be: there's no vector from the original given three ones that is no element of the basis above for S=\mathbb{R}^3 so there's no more to be done.

    Tonio
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  3. #3
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    Red face

    Thanks Tonio I thought I was going mad as I kept coming up with the same answer.
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