A subspace of a vector space $V$ is a subset $H$ of $V$ that has three properties:
a) The zero vector of $V$ is in $H$.
b) $H$ is closed under vector addition. That is for each $u$ and $v$ in $H$, the sum $u+v$ is in $H$.
c) $H$ is closed under multiplication by scalars. That is, for each $u$ in $H$ and each scalar $c$, the vector $cu$ is in $H$.
It would be great if someone could "dumb" this down. It already seems extremely simply, but i'm having a very difficult time applying these.