1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    User Avatar
    Science Advisor
    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).

    Hence....

    Now try 1.
     
  4. Apr 24, 2009 #3
    Thanks for your help, it allowed me to finish the rest of the proof.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Help with a proof involving the span of a subset
  1. Span Proof help ! (Replies: 22)

  2. Spanning and subsets (Replies: 1)

Loading...