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Help with a proof involving the span of a subset

  1. Apr 23, 2009 #1
    1. The problem statement, all variables and given/known data
    Let S be a subset of a vector space V, and let v be an element of V. Show that span(S) = span(S U {v}) if and only if v is an element of span(S)

    2. Relevant equations

    3. The attempt at a solution

    I'm honestly not sure how to get started, I've spent time looking through the text but it hasn't helped.
  2. jcsd
  3. Apr 24, 2009 #2

    matt grime

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    Homework Helper

    You have to do two things.

    1. Suppose that v is in span(S), show that span(S)=span(S u {v}).

    2. Show that if span(S u {v}) = span(S) then v is in span(S).

    Let's look at 2 and break down what you know.

    Definition: span(X) is the space of all linear combinations of objects in X.

    You need to show v is in span(S).

    You know that v is in span(S u {v}) as it is a linear combination of elements of S U {v} trivially.

    You know that span(S u {v}) = span(S).


    Now try 1.
  4. Apr 24, 2009 #3
    Thanks for your help, it allowed me to finish the rest of the proof.
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