Question

Let w be a subset of a vector space v. Prove or disprove: If w w = w, then w is a subspace of v?

Answer 1

To prove or disprove the statement, we need to consider the properties of a subspace. For a subset, w, of a vector space, v, to be a subspace, it must satisfy three conditions: containing the zero vector, closure under addition, and closure under scalar multiplication. The statement that w^2 = w doesn't guarantee that w satisfies these conditions, so it can be disproven with a counterexample.

Step-by-step explanation:

To prove or disprove the statement, we need to consider the properties of a subspace. For a subset, w, of a vector space, v, to be a subspace, it must satisfy three conditions:

1. It must contain the zero vector, 0.
2. It must be closed under addition, meaning that for any two vectors, u and v, in w, the sum of u and v must also be in w.
3. It must be closed under scalar multiplication, meaning that for any vector, u, in w and any scalar, c, the scalar multiple of u, cu, must also be in w.

Now, let's consider the statement that w^2 = w. This implies that every vector in w is the square of another vector in w. However, this doesn't guarantee that w satisfies the three conditions for a subspace. Therefore, we can disprove the statement by providing a counterexample.

For example, let v be the vector space R^2 and let w = {(1, 1)}. In this case, w^2 = {(1, 1)^2} = {(1, 1)}. However, w does not contain the zero vector, so it fails to satisfy the first condition for a subspace. Hence, the statement is disproven.