Let V R2 with the operations u1 u2 v1 v2 u1 v10 and ku1u

Let V = R^2 with the operations (u_1, u_2) + (v_1, v_2) = (u_1 + v_1,0) and k(u_1,u_2) = (ku_1, 0). Determine which vector space axioms are satisfied, and which ones are not.

Solution

There are three axioms for a set being a vector space..

1. Is 0 a member of the set? In this case, \"0\" means (0, 0). Since V being equal to R² contains all (x , y) where x , y R, therefore, (0,0) is in V.

2. If v is in the set, is a*v in the set for any real number a? Let (x, y) be an arbitrary element of V. Then, a(x, y) = ( ax , 0). Which is in V as, when a is a real number and x R. then ax R. ) also belongs to R.
3. If v and w are in the set, is v + w in the set? Let (x₁, y₁) and (x₂, y₂) be in the set V. Then (x₁, y₁) + (x₂, y₂) = (x₁ +x_{2 ,} 0) which is in V when x₁ and x₂ R.

Thus V satisfies all the axioms of a vector space.