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Some technical questions about spans and bases

  1. Jan 6, 2013 #1
    I have some technical questions about spans and bases which my textbook really does not cover very well. I would appreciate answers to them. They are not textbook problems, merely specific quesitons relating to the definitions of "span" and "basis".

    1) If span(S) = V, need the elements of S be in V?
    2) If S is a basis for V, need the elements of S be in V?
    3) If S spans V, does S span every subspace of V?
    4) If S is a basis for V, is S a basis for every subspace of V?
    5) If S is linearly independent, is every subset of S also linearly independent?
    6) If S is linearly dependent, is every superset of S also linearly dependent?

    If I am not wrong, the answers to the questions are:
    1) Yes
    2) Yes
    3) Yes
    4) No
    5) Yes
    6) Yes

    But I would like explicit confirmation.
    All help is appreciated. Thanks.

    BiP
     
  2. jcsd
  3. Jan 6, 2013 #2

    mfb

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    Staff: Mentor

    All correct. For a strict mathematical formulation, you should let everything be a part of a vector space W (can be identical to V, but then questions 1 and 2 are meaningless).

    The span of S contains every subspace of V as subset.
     
  4. Jan 6, 2013 #3

    AlephZero

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    I disagree about (3).

    If S spans V and W is a subspace of V, then the elements of S may not all be in W, so "S spans W" might contradicts the (correct IMO) answer to (1).

    Also the definition of "span" in http://en.wikipedia.org/wiki/Linear_span implies that there is only ONE space that is spanned by a given set S - namely, the intersection of all subspaces of V that contain (all the vectors in) S.
     
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