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    A vector space V

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    Determine if the set R^2 (the real plane) is a vector space with operations defined by the following:

    Addition: (a,b)+(c,d)=(a+c,b+d)

    Scalar Multiplication: k(a,b)=(0,0)

    Prove your answer. If this structure is not a vector space, state all the axioms of a vector space which are not satisfied and explain your answer.

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    Solution Preview

    Solution: Recall the definition of a vector space.

    Definition: A vector space V is a set of elements together with two operations, addition and scalar multiplication, satisfying the following properties:

    Let u,v, and w be vectors in V, and let c and d be scalars.

    (a) u+v is in V. (closure under addition)
    1. u+v=v+u. (commutative law)
    2. u+(v+w)=(u+v)+w. (associative law)
    3. V has a ...

    Solution Summary

    This solution helps proof showing whether or not a structure is a vector space. Axioms of the vector space are stated in in the solution.