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Linear algebra proof help!

  1. Apr 7, 2009 #1
    Let S = {v1,…vn} be a linearly independent set in a vector space V. Suppose w is in V, but w is not in span(S). Prove that the set T = {v1,…vn,w} is a linearly independent set.



    since it says w is in v, does it mean that w is a subspace of V? yet w is not in span(S). I am kinda confused with what it means?


    ANy ideas or thoughts on that?

    Thanks in advance
     
    Last edited: Apr 7, 2009
  2. jcsd
  3. Apr 7, 2009 #2

    matt grime

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    You should make sure you honour capitalizations of words. The lower case w is a vector, V is a vector space. The symbols v1 to vn represent n linearly independent vectors. They span a subspace of V.

    What would it mean if {v1,..,vn,w} were linearly dependent?
     
  4. Apr 7, 2009 #3
    if {v1,..,vn,w} were linearly dependent, it would span V too.
     
  5. Apr 7, 2009 #4

    matt grime

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    You have not been told anything spans V. In fact S cannot span V since you are explicitly told that there is a vector, w, that is in V, and not in the span of S.

    If {v1,..,vn,w} is a linearly dependent set then <INSERT DEFINITION OF LINEARLY DEPENDENT HERE>.
     
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