See subsets in Description below, graph these subsets separately and determine whether either of them is a subspace of V.

Question

Consider as a vector space with the standard operations, and the subsets W₁:={(x, y) ∈ V | xy = 0} and W₂:={(x, y) ∈ V | x + y = 0} of V. Graph these subsets separately and determine whether either of them is a subspace of V.

Answer 1

W₁ is not a subspace of R²:

Example: (3,0) and (0,3) are both in W₁. But (3,0) + (0,3) = (3,3) is not in W₁.

W₂ is a subspace of R²:

Proof:

(1). Let (x, y) and (a, b) be in W₂. Then, x+y = 0 and a+b = 0.

So, (x,y) + (a,b) = (x+a, y+b) which is in W₂ since (x+a) + (y+b) = (x+y) + (a+b) = 0.

Therefore, W₂ is closed under addition.

(2) Let (x,y) be in W₂ and let c be a real number. So, x + y = 0.

c(x,y) = (cx, cy). cx + cy = c(x+y) = c(0) = 0. Thus, c(x,y) is in W₂.

Therefore, W₂ is closed under scalar multiplication.

Since W₂ is closed under both addition and scalar multiplication, W₂ is a subspace of R².