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Linear algebra basis question

  1. Jan 8, 2014 #1
    1. The problem statement, all variables and given/known data
    Let [itex]v_1,...,v_k[/itex] be vectors in a vector space [itex]V[/itex]. If [itex]v_1,...,v_k[/itex] span [itex]V[/itex] and after removing any of the vectors the remaining [itex]k-1[/itex] vectors do not span [itex]V[/itex] then [itex]v_1,...,v_k[/itex] is a basis of [itex]V[/itex]?


    2. Relevant equations



    3. The attempt at a solution
    If [itex]v_1,...,v_k[/itex] span [itex]V[/itex] but [itex]v_1,...,v_{k-1}[/itex] do not then [itex]v_1,...,v_k[/itex] are linearly independent.
    If [itex]v_1,...,v_k[/itex] span [itex]V[/itex] and are linearly independent the [itex]v_1,...,v_k[/itex] is a basis of [itex]V[/itex]
    Is this reasoning correct?
     
  2. jcsd
  3. Jan 8, 2014 #2

    HallsofIvy

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    Yes, your reasoning is correct. If any subset of this set of vectors does not span the vector space, then the original set is independent.
     
  4. Jan 8, 2014 #3
    If I were writing a proof I would want to emphasize that ##v_{k}## is any arbitrary vector of the set and not a named one.

    Personally I would say:
    {##{v_{1}, v_{2}, ... v_{k}}##} \ {##{v_{i}}##} is linearly dependent for all i in {1,2,..,k}.

    But I'm just being nitpicky.
     
  5. Jan 8, 2014 #4

    Office_Shredder

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    This isn't a mathematical point but given the level of the exercise I would guess you are expected to prove this part (but obviously you are the only one who can know what level of detail is required in your homework)
     
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