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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.© BrainMass Inc. brainmass.com March 4, 2021, 11:34 pm ad1c9bdddf
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 ...
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.