Complement (linear algebra)

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In linear algebra, a complement to a subspace of a vector space is another subspace which forms an internal direct sum. Two such spaces are mutually complementary.

Formally, if U is a subspace of V, then W is a complement of U if and only if V is the direct sum of U and W, ${\displaystyle V=U\oplus W}$, that is:

${\displaystyle V=U+W;\,}$

${\displaystyle U\cap W=\{0\}.\,}$

Equivalently, every element of V can be expressed uniquely as a sum of an element of U and an element of W. The complementarity relation is symmetric, that is, if W is a complement of U then U is also a complement of W.

If V is finite-dimensional then for complementary subspaces U, W we have

${\displaystyle \dim V=\dim U+\dim W.\,}$

In general a subspace does not have a unique complement (although the zero subspace and V itself are the unique complements each of the other). However, if V is in addition an inner product space, then there is a unique orthogonal complement

${\displaystyle U^{\perp }=\{v\in V:(v,u)=0{\mbox{ for all }}u\in U\}.\,}$