# Complement (linear algebra)

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

Formally, if *U* and *W* are subspaces of *V*, then *W* is a complement of *U* if and only if *V* is the internal direct sum of *U* and *W*, , that is:

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

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 a finite-dimensional inner product space, then there is a unique *orthogonal complement*

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