13. Sums and intersections of subspaces

Intersection of subspaces

If V and W are subspaces of a space X, then V\cap W is a subspace of X as well.

E.g. if V=\text{lin}((1,0,1,1),(0,1,0,0)), W=\text{lin}((2,0,2,2),(0,0,1,0)), then V\cap W=\text{lin}((1,0,1,1)). If on the other hand V=\{(x,y,z)\colon x+y-z=0\} and W=\{(x,y,z)\colon 2x+2y-z=0\}, then V\cap W is the space of solutions to

    \[\begin{cases} x+y-z=0,\\2x+2y-z=0.\end{cases}\]

Sum of subspaces

If V and W are subspace of X, then V+W=\{v+w\colon v\in V\land w \in W\} is also a subspace of X.

E.g. if V=\text{lin}((1,0,1,1),(0,1,0,0)), W=\text{lin}((2,0,2,2),(0,0,1,0)), then V+ W=\text{lin}((1,0,1,1),(0,1,0,0),(0,0,1,0)).

It is easy to notice that V+W=\text{lin}(V\cup W) and that in the case of finite dimensional spaces \dim(V+W)=\dim V+\dim W-\dim(V\cap W).

Direct sum

We say that X is a direct sum of its subspaces V and W (denoted by X=V\oplus W), if X=V+W, but also V\cap W=\{0\}. In other words, every vector \alpha \in X can be uniquely described as v+w, where v\in V and w\in W. Obviously, if X=V\oplus W, then \dim X=\dim V+\dim W.

E.g. \mathbb{R}^2=\lin((1,0))\oplus\lin((1,1)), but even though \mathbb{R}^2=\lin((1,0),(1,1))+\lin((1,1)), this sum is not direct.