Let V denote the set of all ordered pairs (x, y) and define addition in V as in R2

Question

Let V denote the set of all ordered pairs (x, y) and define addition in V as in R². However, define a new scalar multiplication in V by a (x, y) = (ay, ax). Determine if V is a vector space with these operations.

Answer 1

Hey Honey. I say V is not a vector space. Recall from the axioms of a vector space that we must have

1(x, y) = (x, y) for any (x, y) if V is a vector space. But we see that

1(1, 0) = (1(0), 1(1)) = (0, 1) ≠ (1, 0).

Thus V is not a vector space.

Hope this helps!