Difference between revisions of "Complement (linear algebra)"

From Citizendium
Jump to: navigation, search
(New entry, just a stub)
 
(subpages)
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''.
  

Revision as of 20:19, 28 November 2008

This article is a stub and thus not approved.
Main Article
Discussion
Related Articles  [?]
Bibliography  [?]
External Links  [?]
Citable Version  [?]
 
This editable Main Article is under development and subject to a disclaimer.

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:

Clearly this 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