A subspace of a vector space 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.

If your original vector space was $V=\mathbb R^3$, then the possible subspaces are:


*

*The whole space

*Any plane that passes through $0$

*Any line through $0$

*The singleton set, $\{0\}$


One reading for the definition is that $H$ is a subspace of $V$ if it is a sub-set of $V$ and it is also a vector space under the same operations as in $V$.
