Question

Let H be a set that represents the set of points on and inside an in the xy-plane. Find two specific examples of two vectors, v and w, and a vector u and a scalar c to show that H is not a subspace of R².

Answer 1

To show that H is not a subspace of R², we can find examples that violate the subspace properties.

Step-by-step explanation:

To show that H is not a subspace of R², we need to find two specific examples that violate one or more of the subspace properties:

1. Closed under addition: We can choose two vectors v and w such that their sum v + w lies outside H. For example, let v = (1, 1) and w = (-1, -1). The sum v + w = (1 - 1, 1 - 1) = (0, 0) lies on the boundary of H and not inside it.
2. Closed under scalar multiplication: We can choose a scalar c and a vector u such that c*u lies outside H. For example, let c = -1 and u = (2, 2). The scalar multiple c*u = (-2, -2) lies outside H.

Therefore, H is not a subspace of R² since it violates one or more of the subspace properties.