An ordered basis for V is said to be adapted to a flag V₀ ⊂ V₁ ⊂ ... ⊂ V_k 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 entry and 0's elsewhere. Concretely, the standard flag is the sequence of subspaces:

${\displaystyle 0<\left\langle e_{1}\right\rangle <\left\langle e_{1},e_{2}\right\rangle <\cdots <\left\langle e_{1},\ldots ,e_{n}\right\rangle =K^{n}.}$ 

An adapted basis is almost never unique (the counterexamples are trivial); see below.

A complete flag on an inner product space has an essentially unique orthonormal basis: it is unique up to multiplying each vector by a unit (scalar of unit length, e.g. 1, −1, i). Such a basis can be constructed using the Gram-Schmidt process. The uniqueness up to units follows inductively, by noting that ${\displaystyle v_{i}}$  lies in the one-dimensional space ${\displaystyle V_{i-1}^{\perp }\cap V_{i}}$ .

More abstractly, it is unique up to an action of the maximal torus: the flag corresponds to the Borel group, and the inner product corresponds to the maximal compact subgroup.^[2]

The stabilizer subgroup of the standard flag is the group of invertible upper triangular matrices.

More generally, the stabilizer of a flag (the linear operators on V such that ${\displaystyle T(V_{i})<V_{i}}$  for all i) is, in matrix terms, the algebra of block upper triangular matrices (with respect to an adapted basis), where the block sizes ${\displaystyle d_{i}-d_{i-1}}$ . The stabilizer subgroup of a complete flag is the set of invertible upper triangular matrices with respect to any basis adapted to the flag. The subgroup of lower triangular matrices with respect to such a basis depends on that basis, and can therefore not be characterized in terms of the flag only.

The stabilizer subgroup of any complete flag is a Borel subgroup (of the general linear group), and the stabilizer of any partial flags is a parabolic subgroup.

The stabilizer subgroup of a flag acts simply transitively on adapted bases for the flag, and thus these are not unique unless the stabilizer is trivial. That is a very exceptional circumstance: it happens only for a vector space of dimension 0, or for a vector space over ${\displaystyle \mathbf {F} _{2}}$  of dimension 1 (precisely the cases where only one basis exists, independently of any flag).

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.

From the point of view of the field with one element, a set can be seen as a vector space over the field with one element: this formalizes various analogies between Coxeter groups and algebraic groups.

Under this correspondence, an ordering on a set corresponds to a maximal flag: an ordering is equivalent to a maximal filtration of a set. For instance, the filtration (flag) ${\displaystyle \{0\}\subset \{0,1\}\subset \{0,1,2\}}$  corresponds to the ordering ${\displaystyle (0,1,2)}$ .

• Filtration (mathematics)
• Flag manifold
• Grassmannian
• Matroid

• Shafarevich, I. R.; A. O. Remizov (2012). Linear Algebra and Geometry. Springer. ISBN 978-3-642-30993-9.