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Math Help - Proof relating to Vector Spaces

  1. #1
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    Proof relating to Vector Spaces

    Let  V be the set of all pairs  (x,y) or real numbers and  \Re be the feild of real numbers. Define  (x,y) +  (x_{1},y_{1}) = (x+x_{1},y+y_{1}), c \cdot (x,y) = (cx,y) . Prove that  V(\Re) is NOT a vector Space.

    I can't seem to figure out which of the 10 conditions of Vector Spaces is being violated. My guess is that they involve the ones relating to the external composition since  V clearly is an abelian group, but I cannot disprove any of the conditions. Please help!
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  2. #2
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    Re: Proof relating to Vector Spaces

    The definition of addition is standard, so the problem is not there. There are only four axioms related to scalar multiplication (i.e., multiplication by a scalar). Double-check them carefully by writing everything explicitly. Finding the failing axiom is not hard.
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  3. #3
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    Re: Proof relating to Vector Spaces

    Quote Originally Posted by sashikanth View Post
    Let  V be the set of all pairs  (x,y) or real numbers and  \Re be the feild of real numbers. Define  (x,y) +  (x_{1},y_{1}) = (x+x_{1},y+y_{1}), c \cdot (x,y) = (cx,y) . Prove that  V(\Re) is NOT a vector Space.
    This is an addendum to reply #2.
    It is possible that the way this is done will depend upon the set of axioms and definitions in use.
    The linear algebra textbook by Larry Smith here Proposition 2.1: 0\mathcal{A}=0. for any vector \mathcal{A}.
    Note that is a theorem not an axiom.

    In this case what is 0(1,1)=~?.
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