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Linear independent

  1. Aug 15, 2010 #1
    1. The problem statement, all variables and given/known data

    If v11,.....,vk,v are linear independent, prove that v[tex]\notin[/tex]

    [tex]\left\langle[/tex]v1,......,vk[tex]\right\rangle[/tex]


    2. Relevant equations

    n/a

    3. The attempt at a solution

    i can prove it by contrapositive, but i'm curious how to proof it with

    "If v11,.....,vk,v are linear independent" in the beginning,

    any idea? T_T
     
    Last edited: Aug 15, 2010
  2. jcsd
  3. Aug 15, 2010 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    I really cannot read what you have here.

    Are you trying to show that "if [itex]\{v_1, v_2, \cdot\cdot\cdot, v_k, v\}[/itex] is linearly independent, then v is not in the span of [itex]\{v_1, v_2, \cdot\cdot\cdot, v_k\}"?

    Since proof by contradiction works nicely, why look for a "direct" proof?
     
  4. Aug 15, 2010 #3
    i'm just curious, maybe there is a way that i don't know,

    anyway, you wrote

    "[itex]
    \{v_1, v_2, \cdot\cdot\cdot, v_k, v\}
    [/itex] is linearly independent"

    is it the same thing as "[itex]
    v_1, v_2, \cdot\cdot\cdot, v_k, v
    [/itex] are linear independent" ??
     
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