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,  V = U ⊕ W {\displaystyle V=U\oplus W} , that is:

V = U + W ; {\displaystyle V=U+W;\,}
U ∩ W = { 0 } . {\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

dim ⁡ V = dim ⁡ U + dim ⁡ W . {\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

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