Flag (linear algebra)

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 V_i = d_i 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 d_i = 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 d_i basis vectors form a basis for V_i 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 V_i be the span of the first i basis vectors. For example, the standard flag in Rⁿ is induced from the standard basis {e₁, …, e_n} where e_i 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.

See also

• flag manifold