# 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.

Addition:

(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.