What is Direct Sum Decomposition

Question

Suppose that $V$ is a finite dimension inner product space and $W$ is a subspace of $V$. Then we know that $V = W \oplus W^{\perp}$.

What is this $\oplus$ operator? Is it equivalent to union $\bigcup$ or intersection $\bigcap$?

Answer 1

Let $V$ a vector space and let $U_1,\,U_2$ subspaces of $V$.

You may already know that $W=U_1+ U_2=\{ u_1+u_2\,\,:\,\,u_1\in U_1,\,u_2\in U_2\}$ is a subspace of $V$. But if $U_1\cap U_2=\{ 0\}$ where $0$ is the identity element in the abelian group $(V,+)$, then $W$ is called the direct sumof $U_1$ and $U_2$ and we write: $W=U_1 \oplus U_2$. An equivalent definition is that $W=U_1\oplus U_2$ if and only if $\forall w\in W,\exists !(u_1,u_2)\in U_1\times U_2,\,w=u_1+u_2$.

To compare $\oplus$ with operations in sets, let's say that $U_1\oplus U_2$ is equivalent to $A\cup B$ when $A\cap B=\emptyset$. $U_1$ and $U_2$ are subspaces of $V$ so they must have $0$, thus we take instead $U_1\cap U_2=\{0\}$.

We can prove that if $V$ is a finite-dimensional vector space and $U_1$ and $U_2$ are subspaces of $V$ (thus they're both finite-dimensional) then $V=U_1\oplus U_2$ if and only if $U_1\cap U_2=\{0\}$ and $\dim V=\dim U_1+\dim U_2$.

More generally, suppose $U_1,...,U_n$ are finite-dimensional subspaces of a vector space $V$ such as $U_1+...+U_n$ is a direct sum. Then $U_1\oplus ...\oplus U_n$ is finite-dimensionale and we have $\dim (U_1\oplus ...\oplus U_n)=\dim U_1+...+\dim U_n$.

So when you say $V=W\oplus W^\perp$ it means that $\forall v\in V,\exists !(x,y)\in W\times W^\perp,\,v=x+y$.