A nonempty subset W of a vector space V is a subspace of V i

A nonempty subset W of a vector space V is a subspace of V if and only if the following two conditions are satisfied for any vectors u, v W and any scalar c R: If u and v are vectors in W, then the vector u + v is also in W. If u is a vector in W and c is a scalar, then the vector cu is also in W. Are the following sets of vectors W, subspaces of V? Give either a proof (if you think the set is a subspace) or a counterexample (if you think it is not). Counterexamples must demonstrate which subspace property given above they violate. V = R^3 W = { [alpha beta gamma] R^3 such that beta Greaterthanorequalto 0} V = R^3 W = {[alpha beta gamma] R^3 such that alpha beta Greaterthanorequalto 0}

Solution

(a) The system W is not a subspace in R3

Example: consider the case [2 1 0], then it belongs to W, since the contraints value of beta is greater than zero

But if we multiply it with a constant -1, then we get the R3 vectors as [-2 -1 0], but in this case beta is not greater than zero, hence given R3 vector doesn\'t belong to W

Therefore, W is not a subspace of V

(b)

The system W is not a subspace of R3

Consider w1 = [2 4 0], w2 = [-3 -3 0]

In both w1 and w2 alpha*beta is greater than zero, since for w1 it is 8 and w2 it is 9

but the addition w1+w2 will be [-1 1 0], in this case alpha*beta = -1

which doesn\'t belong to W, hence W is not closed

Therefore W is not the subspace of R3