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Math Help - Proof

  1. #1
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    Question Proof

    Let V be a vector space, v be a vector in V and c be scalar. Prove that

    If cv=0, then either c=0 or v=0
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  2. #2
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    Quote Originally Posted by lm6485 View Post
    Let V be a vector space, v be a vector in V and c be scalar. Prove that

    If cv=0, then either c=0 or v=0

    Apply directly the axioms of vector space: if c is a non zero scalar then it has an inverse scalar, so...

    Tonio
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  3. #3
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    I have no idea where to go with this. If c is a non zero scalar then what axiom would I use to prove that v=0?
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  4. #4
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    If c is a nonzero scalar, then c^{-1} exists. Multiply both sides of cv=0 by c^{-1}.
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  5. #5
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    Quote Originally Posted by lm6485 View Post
    Let V be a vector space, v be a vector in V and c be scalar. Prove that

    If cv=0, then either c=0 or v=0
    I found this:

    Suppose that
    cv = 0 and c (not =) 0. We must show that v = 0. Now there exists an element c^-1

    of
    K satisfying c^-1 c = 1, since any non-zero element of a field has a multiplicative inverse. It then follows from the vector space axioms and (Let V be a vector space over a field K. Then c0 = 0 and 0v = 0 for all elements c of K and elements v of V . )that

    v = 1v = (c^-1c)v = c^-1(cv) = c^-10 = 0, as required.
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