Which of the following sets W are subspaces of V Justify you

Which of the following sets W are subspaces of V? Justify your answers with an argument of why it is closed under addition and scalar multiplication or a counterexample showing it is not. V = R^3 and W = {x = [x_1 x_2 x_3] R^3: x_1 greaterthanorequalto 0 and x_3 = 0} V = R^3 and W = R^2.

Solution

i.

NOt a subspace

Consider a vector in W

[1,2,0]^T

Multiplying by scalar -1 gives

[-1,-2,0]^T which is not in W

Hence not closed under scalar multiplication

Hence not a subspace

ii)

W is not a subspace of R3 because it is not a subset of R3

Elements of W are of the form: (x,y)

But elements of R3 are of the form: (x,y,z)

So, W is not a subset of R3 and hence not a subspace