Some interesting linear algebra theories

if $U$ and $W$ are subspaces for a vector space $V$ then $U \cap W$ is a subspace (of $V$ ).
Definition: $\{U + W\}$ denotes the set of all the vectors possible to construct by adding a vector from $U$ with a vector from $W$ where $U$ and $W$ may be either set of vectors or subspace of vectors.
If $U$ and $W$ are subspaces for a vector space $V$ then $\{U + W\}$ is the smallest subspace which contain both $U$ and $W$ .