Direct sum of subspaces - Imported Knowlpedia Content

Let $X$ be a vector space and let $M,N$ be linear subspaces.

Let $Y:=M+N$ be their (set) sum. We say that $Y$ is the direct sum of $M$ and $N$ if

M\cap N=\{0\}.

In this case we write

Y=M\oplus N.

Direct sums formalize "adding independent directions": if $M\cap N=\{0\}$ , then no nonzero vector lies in both subspaces. The key structural property is uniqueness of decomposition, captured in the direct sum characterization.

Examples

In $\mathbb{R}^2$ , let $M=\operatorname{span}\{(1,0)\}$ and $N=\operatorname{span}\{(0,1)\}$ . Then $\mathbb{R}^2=M\oplus N$ .
In $\mathbb{R}^3$ , the $xy$ -plane and the $z$ -axis form a direct sum: $\mathbb{R}^3=\mathbb{R}^2\times\{0\}\ \oplus\ \operatorname{span}\{(0,0,1)\}$ .
If $M=N\neq\{0\}$ , then $M+N=M$ but $M\cap N=M\neq\{0\}$ , so this is not a direct sum.