Consider the vector space Vropf2 over the set of scalars rop

Consider the vector space V=ropf^2 over the set of scalars ropf. Define U={(x, y) in ropf^2 such that x greaterthanorequalto 0, y greaterthanorequalto 0}. Is U a subspace of ropf^2?

Solution

We know that (x,y ) = (x, y) , where is an arbitrary scalar. Since need not be 0, therefore when < 0 , x and y are both 0. Thus, U is not a sub-space of R² since it is not closed under scalar multiplication.