Let $V$ be a vector space, and let $W$ be a subset of $V$. We define $W$ to be a subspace if $W$ satisfies the following conditions: - If $\boldsymbol v, \boldsymbol w$ are elements of $W$, their sum $v+w$ is also an element of $W.$ - If $\boldsymbol v$ is an element of $W$ and $c$ a number, then $c \boldsymbol v$ is an element of $W.$ - The element $O$ of $V$ is also an element of $W.$