Explain why the following sets are not vector subspaces of the indicated vector spaces. W = {(x, y) ∈R^2 |x, y ∈Q}.

Answer 1

One of the requirements for a subset of a vector space to be a subspace of the vector space is that the subspace must be closed under scalar multiplication. In other words, if v is in the subset then cv must also be in the subset (c is any real number).

W is not a subspace of lR² because W is not closed under scalar multiplication.

For example, let v = (1/5, 1/6) and let c = √2. v is in W but cv is not.

Therefore, W is NOT a subspace of lR².

Answer 2