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Homework Help: 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


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    Science Advisor

    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


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    Staff Emeritus
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    Gold Member

    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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