November 26th, 2012, 04:08 AM  #1 
Newbie Joined: Nov 2012 Posts: 3 Thanks: 0  Vector Spaces anyone?
Let V = Real Number. Define ? + ? = 2? ? and c(?) = c?. For example, 3+5=2(3)5=1 and 3(5)=15. Is V a vector space or not? Support your answer.

November 26th, 2012, 04:54 AM  #2 
Math Team Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 233  Re: Vector Spaces anyone?
For it to be a vector space it must satisfy ALL the vector space axioms. [attachment=0:1rpwmkpu]Vector space  Wikipedia, the free encyclopedia.png[/attachment:1rpwmkpu] I will show you how it fails the first axiom, thereby disqualifying this particular relation being a vector space. axiom 1) U + (V + W) = (U + V) + W let U = 2 , V = 3 , W = 4 2 + (3 + 4) = 2 + [2(3)  4] = 2 + (64) = 2 + 2 = 2(2)  2 = 2 (2 + 3) + 4 = [2(2) 3] + 4 = (4  3) + 4 = 1 + 4 = 2(1)  4 = 2 so the relation is not associative under the operation of addition, we didn't get the same result. It actually fails 4 axioms, can you find the other 3? 

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