Let VR3 and Wabc such that a2b2c2 Is W a subspace of V If so

Let V=R3 and W={(a,b,c) such that a2+b2=c2}. Is W a subspace of V? If so what is its dimension?

Solution

it is not a subspace because it is not closed under addition

for example (1,0,1) and (0,1,1) are in W since 1^2+0^2=0^2+1^2=1^2 but (1,0,1)+(0,1,1)=(1,1,2) is not in W since 1^2+1^2 is different than 2^2