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Math Help - vector space proofs

  1. #1
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    vector space proofs

    how do you prove that if V is a vector space and v is a vector and c is any scalar, then if cv =0, it means that c=0 or v=0?

    i was thinking of using the fact that since v is in V, there exist a negative vector -v in V.

    so cv + (-cv) = c ( v+ (-1)v) =c0 = 0

    then after that, im not sure how im going to show that c=0 or v=0..
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by alexandrabel90 View Post
    how do you prove that if V is a vector space and v is a vector and c is any scalar, then if cv =0, it means that c=0 or v=0?

    i was thinking of using the fact that since v is in V, there exist a negative vector -v in V.

    so cv + (-cv) = c ( v+ (-1)v) =c0 = 0

    then after that, im not sure how im going to show that c=0 or v=0..
    Let \mathbf{v}=\left<v_1,v_2,\dots,v_n\right>\in V.

    Now, c\mathbf{v}=\mathbf{0}\implies c\left<v_1,v_2,\dots,v_n\right>=\left<0,0,\dots,0\  right>\implies\left<cv_1,cv_2,\dots,cv_n\right>=\l  eft<0,0,\dots,0\right>.

    This implies that cv_i=0,\,1\leq i\leq n.

    By the zero product property, it follows that if cv_i=0, then c=0 or v_i=0\implies \mathbf{v}=\mathbf{0}.

    Does this make sense?
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  3. #3
    MHF Contributor Bruno J.'s Avatar
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    That is not bad, Chris, but you assume that the vector space has a finite basis. Some vector spaces have infinite bases, sometimes even uncountable bases, and then the argument is hard to generalize.

    Suppose cv=0. If c=0, we are done. If c \neq 0, c is invertible, and hence c^{-1}cv=0 \Rightarrow v = 0 (We have used the axiom which states that a(bv)=(ab)v for any a, b in the field).

    Hence if cv=0, v=0 or c=0.
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