Consider the set of vectors xy should display vertically in

Consider the set of vectors (x,y) (should display vertically) in the XY plane, such that y < and = to 0. Give two different reasons to show that this set is not a vector space

Solution

Let X = ( x, y)^T be an arbitrary element of the set S = { (x, y)^T : x , y R and y 0 } . Also, let be an arbitrary scalar. Then X = ( x, y)^T = (x, y)^T does not belong to S when is negative ( as , then y 0). Thus S is not closed under scalar multiplication. Now, every vector space has a basis. Therefore, if X and X₁ = (x₁ , y₁)^T are two elements in the basis of S,  and if and are two arbitrary scalars , then  X + X₁ need not belong to S ( as is the case when and are both negative).Thus S does not have a basis . Hence S is not a vector space.