Linear algebra basis question

  • #1
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Homework Statement


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]?


Homework Equations





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?
 

Answers and Replies

  • #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.
 
  • #3
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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.
 
  • #4
Office_Shredder
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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.

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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