Direct sum ($\oplus$)

The direct sum $U\oplus W$ of two vector spaces consists of pairs $(u,w)$ with componentwise operations. A vector space is an internal direct sum of subspaces $U$ and $W$ exactly when every vector has a unique expression $u+w$ with $u\in U$ and $w\in W$.

For bases $(u_i)$ of $U$ and $(v_j)$ of $W$, the displayed pairs form a basis of $U\oplus W$. Expanding each component proves spanning; comparing components of a vanishing linear combination proves linear independence. Therefore $\dim (U\oplus W)=\dim U+\dim W$ for two finite-dimensional vector spaces.

If $V=A_1\oplus A_2=A_2\oplus A_3=A_1\oplus A_3$ with nonzero subspaces, choose $0\ne v\in A_3$ and write $v=a_1+a_2$ using the first direct sum of vector spaces. Both components are nonzero. The plane spanned by them meets $A_1$ and $A_2$ in their component lines and meets $A_3$ in the line through $v$. Its intersection with $A_3$ cannot be the whole plane, since $a_1\notin A_3$.

An orthogonal direct sum is a direct sum whose distinct summands are mutually orthogonal. For $v\in V$ and $w\in W$, the Pythagorean identity gives $\Vert v+w{\Vert }^2=\Vert v{\Vert }^2+\Vert w{\Vert }^2$. In a Hilbert space, a closed subspace of a Hilbert space and its orthogonal complement give an orthogonal direct sum of the entire space. The component maps are orthogonal projections and have operator norm one when their ranges are nonzero.

A submodule $N$ of a module $M$ is a direct summand if some submodule $L$ gives $M=N\oplus L$. Equivalently, the inclusion has a left inverse R-module homomorphism, or $N$ is the image of an idempotent endomorphism of $M$.

Integer row vectors forming a square unimodular matrix are a basis of the corresponding free abelian group: the adjugate formula makes the inverse integral. The subgroup spanned by any subset of that basis is therefore a direct summand, complemented by the remaining basis vectors. In contrast, a proper finite-index subgroup of a free abelian group cannot be a direct summand: its nonzero finite quotient would have to embed as a subgroup of a torsion-free group.

For finite-dimensional subspaces $U,W$ of a common vector space,

For every subspace $U$ of a finite-dimensional vector space $V$, extending a basis of $U$ to a basis of $V$ produces a subspace $W$ such that $V=U\oplus W$.