Exercise 12

Example 2.16 says that for any vector $\vec{v}$ that is an element of a vector space $V$, the set $\{\hskip 1.00006ptr\cdot\vec{v}\mid r\in\mathbb{R}\hskip 1.00006pt\}$ is a subspace of $V$. (This is simply the span of the singleton set $\{\hskip 1.00006pt\vec{v}\hskip 1.00006pt\}$.) Must any such subspace be a proper subspace?