Is the subset W x y z y 0 R3 a vector subspace of R3 Expl

Is the subset W = {(x, y, z) | y 0} R^3 a vector subspace of R^3 ? Explain why or why not.

Is the subset W = {(x, y, z) | y = 0} R^3 a vector subspace of R^3 ? Explain why or why not.

Solution

1. y >= 0

Let us take a scalar c such that c < 0.

Now c.y <= 0.

=> cy does not belongs to W.

=> W is not the vector subspace of R³ as we know that y is element of W such that y>=0 if W is a vector subspace of R³ then for any scalar c, cy must be belongs to W which can not be seen in this problem.

2. y = 0

Let us take any scalar c.

cy = 0, for any c.

Let us take x = 0, z = 0

x + y + z = 0 + 0 + 0 = 0

=> 0 vector belongs to W.

Also x + y + z belongs to W.

Therefore we can say W is the vector space of R³.