I'm asked to verify if:
$B=(x,y,z) \in \mathbb{R}^3 : ||(x,y,z)|| \leq 1$
is a subspace. I know I have to check for the zero vector, addition and scalar multiplication. Here lies my question however.
If I use $(0,0,0)$ to check if the zero vector exists, it clearly does. However, if I multiply $(0,0,1)$ by some scalar, say, $50$, scalar multiplication does not hold so this isn't a subspace.
Here lies my problem. I thought because the $0$ vector exists, this set MUST be a subspace and hence, scalar multiplication and addition should hold.
Am I wrong in assuming that if $0$ vector exists, all the other axioms must hold? Or is it possible that some axioms work and some fail?