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Proving linear dependence (2.0)

  1. Nov 17, 2011 #1
    I have to prove:

    Consider [itex]V=F^{n}[/itex]. Let [itex]\mathbf{v}\in V/\{e_{1},e_{2},...,e_{n}\}[/itex]. Prove [itex]\{e_{1},e_{2},...,e_{n},\mathbf{v}\}[/itex] is a linearly dependent set.

    My attempts at a proof:

    Since [itex]{e_{1},e_{2},...,e_{n}}[/itex] is a basis, it is a linearly independent spanning set. Therefore, any vector [itex]\mathbf{v}\in V[/itex] can be written as a linear combination of [itex]{e_{1},e_{2},...,e_{n}}[/itex]. Therefore, the set [itex]\{e_{1},e_{2},...,e_{n},\mathbf{v}\}[/itex] with [itex]\mathbf{v}\in V[/itex] must be linearly dependent.

    Am I on the right track?
     
  2. jcsd
  3. Nov 17, 2011 #2
    Yes, that's ok.
     
  4. Nov 17, 2011 #3

    Deveno

    User Avatar
    Science Advisor

    because...if we write:

    v = v1e1+v2e2+...+vnen,


    then: v1e1+v2e2+...+vnen - 1v

    is a linear combination of {e1,e2,...,en,v} that sums to the 0-vector, and yet not all the coefficients in this sum are 0 (the one for v is -1).
     
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