Direct sum of subspaces

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

In this case we write

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.