Flag (linear algebra)
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In mathematics, particularly in linear algebra, a flag is an increasing sequence of subspaces of a vector space V. Here "increasing" means each is a proper subspace of the next (see filtration):
If we write the dim Vi = di then we have
where n is the dimension of V (assumed to be finite-dimensional). Hence, we must have k ≤ n. A flag is called a complete flag if di = i, otherwise it is called a partial flag.
A partial flag can be obtained from a complete flag by deleting some of the subspaces. Conversely, any partial flag can be completed (in many different ways) by inserting suitable subspaces.
The signature of the flag is the sequence d0, d1, … dk.
[edit] Bases
An ordered basis for V is said to be adapted to a flag if the first di basis vectors form a basis for Vi for each 0 ≤ i ≤ k. Standard arguments from linear algebra can show that any flag has an adapted basis.
Any ordered basis gives rise to a complete flag by letting the Vi be the span of the first i basis vectors. For example, the standard flag in Rn is induced from the standard basis {e1, …, en} where ei denotes the vector with a 1 in the ith slot and 0's elsewhere.
[edit] Subspace nest
In an infinite-dimensional space V, as used in functional analysis, the flag idea generalises to a subspace nest, namely a collection of subspaces of V that is a total order for inclusion and which further is closed under arbitrary intersections and closed linear spans. See nest algebra.
