Let $V$ be any vector space. Let $c$ be in $\mathbb{R}$ and $\mathbf{x}$ be in $V$. Prove the following.

(a) If $c \ne 0$ and $c \mathbf{x} = \mathbf{0}$, then $\mathbf{x} = \mathbf{0}$.


(b) If $\mathbf{x} \ne \mathbf{0}$ and $c \mathbf{x} = \mathbf{0}$, then $c = 0$.