Let V be the set of all ordered pairs of real numbers, with addition being defined as:
[itex](x_1 , x_2 ) + (y_1 , y_2 ) = (x_1 + y_1 , x_2 + y_2 )[/itex]
and scalar multiplication defined as:
[itex]\alpha \circ (x_1 , x_2 ) = (\alpha x_1 , x_2)[/itex]
Is V a vector space with these operations? Justify your answer.
The Attempt at a Solution
I am thinking yes, because the scalar multiplication rule does not seem to violate any of the 8 axioms for vector spaces, but it seems wrong intuitively.