Let S n 0 n is an integer Is S a subspace of R2 If your ans

Let S = {(n, 0): n is an integer}. Is S a subspace of R^2? If your answer is \"yes\" then prove that S is a subspace of R^2. If your answer is \"no\" then give a vector space axiom that does not hold and a specific example where the axiom does not hold.

Solution

1.

No

Because it is not closed under scalar multiplication

Consider: (1,0) in S

Multiply by scalar :0.5 gives

(0.5,0) which is not in S

Hence not closed under scalar multiplication

Hence not a subspace