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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:

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: 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.