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):
- <math>\{0\} = V_0 \sub V_1 \sub V_2 \sub \cdots \sub V_k = V.<math>
If we write the dim Vi = di then we have
- <math>0 = d_0 < d_1 < d_2 < \cdots < d_k = n.<math>
Where n is the dimension of V (assumed to be finite-dimensional). Note that 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 be deleting some of the subspaces. Conversely, any partial flag can be completed (in many different ways) by inserting suitable subspaces.
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.
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, nest algebra.