Let V be a finite-dimensional vector spaceMathworldPlanetmath. A filtrationMathworldPlanetmathPlanetmath of subspacesPlanetmathPlanetmath


is called a flag in V. We speak of a complete flag when


for each i=1,,n.

Next, putting


we say that a list of vectors (u1,,udn) is an adapted basis relative to the flag, if the first d1 vectors give a basis of V1, the first d2 vectors give a basis of V2, etc. Thus, an alternate characterization of a complete flag, is that the first k elements of an adapted basis are a basis of Vk.


Let us consider n. For each k=1,,n let Vk be the span of e1,,ek, where ej denotes the jth basic vector, i.e. the column vectorMathworldPlanetmath with 1 in the jth position and zeros everywhere else. The Vk give a complete flag in n . The list (e1,e2,,en) is an adapted basis relative to this flag, but the list (e2,e1,,en) is not.


More generally, a flag can be defined as a maximal chain in a partially ordered setMathworldPlanetmath. If one considers the poset consisting of subspaces of a (finite dimensional) vector space, one recovers the definition given above.

Title flag
Canonical name Flag
Date of creation 2013-03-22 12:42:35
Last modified on 2013-03-22 12:42:35
Owner rmilson (146)
Last modified by rmilson (146)
Numerical id 9
Author rmilson (146)
Entry type Definition
Classification msc 06A06
Classification msc 15A03
Defines adapted basis
Defines complete flag