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Math Help - spanning sets

  1. #1
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    Oct 2008
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    spanning sets

    Take and give an example to show that it is possible that Span(S) = Span (S') even though and .

    i having problems with this question as i know that i am looking for a set of vectors say S which are also in S' but in S' there has to be more vectors than in S in order for S' to be satisfied and not equal to S.Finding vectors is what im finding hard to grasp

    if anyone could help it wolud be much apreciated trying to do it for months
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  2. #2
    Senior Member
    Joined
    Feb 2008
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    This is the case whenever S'=S\cup T, where every vector of T is a linear combination of the vectors of S.

    For example, let:

    S=\left\{\begin{array}{ccc}\left[\begin{array}{c}1\\0\\0\end{array}\right]&\left[\begin{array}{c}0\\1\\0\end{array}\right]&\left[\begin{array}{c}0\\0\\1\end{array}\right]\end{array}\right\}

    such that span(S)=\mathbb{R}; then let:

    T=\left\{\begin{array}{cc}\left[\begin{array}{c}1\\1\\0\end{array}\right]&\left[\begin{array}{c}0\\1\\1\end{array}\right]\end{array}\right\}

    which means T\cup S=S', and:

    S'=\left\{\begin{array}{ccccc}\left[\begin{array}{c}1\\0\\0\end{array}\right]&\left[\begin{array}{c}0\\1\\0\end{array}\right]&\left[\begin{array}{c}0\\0\\1\end{array}\right]&\left[\begin{array}{c}1\\1\\0\end{array}\right]&\left[\begin{array}{c}0\\1\\1\end{array}\right]\end{array}\right\}

    which makes span(S')=\mathbb{R}=span(S).
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