Intersection of subspaces
If and are subspaces of a space , then is a subspace of as well.
E.g. if , , then . If on the other hand and , then is the space of solutions to
Sum of subspaces
If and are subspace of , then is also a subspace of .
E.g. if , , then .
It is easy to notice that and that in the case of finite dimensional spaces .
We say that is a direct sum of its subspaces and (denoted by ), if , but also . In other words, every vector can be uniquely described as , where and . Obviously, if , then .
E.g. , but even though , this sum is not direct.