# Difference between revisions of "Complement (linear algebra)"

(New entry, just a stub) |
(uniqueness of expression) |
||

(One intermediate revision by the same user not shown) | |||

Line 1: | Line 1: | ||

+ | {{subpages}} | ||

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''. | 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''. | ||

Line 6: | Line 7: | ||

:<math>U \cap W = \{0\} .\,</math> | :<math>U \cap W = \{0\} .\,</math> | ||

− | + | 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 | If ''V'' is finite-dimensional then for complementary subspaces ''U'', ''W'' we have |

## Latest revision as of 20:17, 12 December 2008

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*, , 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 an inner product space, then there is a unique *orthogonal complement*