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For x, y in a vector space V, c in F, if cx=0 then c=0?

  1. Nov 18, 2011 #1
    1. The problem statement, all variables and given/known data

    For x, y in a vector space V, c in F, if cx=0 then c=0?
    How do you prove this?
    This is originally from Friedman, Linear Algebra, p.12.
    To prove this, I can use a few facts:

    (1) cancellation law
    (2) 0, -x are unique
    (3) 0x = 0

    with basic vector space definition.

    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Nov 18, 2011 #2

    I like Serena

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    Welcome to PF, julypraise! :smile:

    Your proposition is only true for non-zero x.
    Should I assume that is an additional constraint?

    If so, then to prove it, I suggest the following approach.

    Start with: suppose that c≠0, and then multiply with the inverse of c.
     
  4. Nov 19, 2011 #3
    Okay, thanks. Let me follow your instruction.

    Prop. For x in V not equal to 0, c in F, if cx=0 then c=0.

    Proof. Suppose c is not equal 0. Then if we times 1/c on the both sides of cx=0
    we get 1x = c0 = 0 Therefore x = 0, which is contradiction. QED

    Okay it works fairly well. Thx.
     
  5. Nov 19, 2011 #4

    I like Serena

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    Good!

    Actually you get: c-1cx=c-10, therefore 1x=x=0. This is use of the cancellation law.
     
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