Let V-R^2 with the standard scalar multiplication and addition + · defined by: (x1,y1)+·(x2,y2)=(x1+x2,y1-y2). Show that this is NOT a vector space.
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Since each element `v in R^2` has standard scalar multiplication, to prove that this is not a vector space, we need to focus on the vector space axioms that relate to the vector space axioms of addition.
To prove that this is not a vector space, we just need to find a single example where one of the rules fails. In particular, consider the rule for associativity:
and consider the three vectors:
Now the left side of the associativity rule gives:
On the other hand, the right side of the associativity rule gives:
Since the two sides are not equal, then the associativity rule for vector spaces fails.
This is not a vector space.
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