Flag

(from Wikipedia)
A flag is an increasing sequence of subspaces of a finite-dimensional vector space $v$ . Here "increasing" means each is a proper subspace of the next:

{0} = V_{0} \subset V_{1} \subset V_{2} \subset \dots \subset V_{k} = V

Etymology: suppose I have an actual flag in the sense of a piece of fabric, and I want to explain to someone how to attach it properly to a flagpole. First I need to say which of the four sides should be attached to the flagpole, lest the flag be flown sideways or backwards. Then, on that distinguished side, I need to specify one corner as the top corner, lest the flag be flown upside down. So we have a rectangle with a distinguished edge, and the distinguished edge has a distinguished endpoint.

Bases

An ordered basis for V is said to be adapted to a flag _V_0 ⊂ V_1 ⊂ ... ⊂ 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 $(e_{1}, . . ., e_{n})$ where $e_{i}$ denotes the vector with a 1 in the $i$ th entry and 0's elsewhere. Concretely, the standard flag is the sequence of subspaces:

0 < ⟨ e_{1} ⟩ < ⟨ e_{1}, e_{2} ⟩ < \dots < ⟨ e_{1}, \dots, e_{n} ⟩ = K^{n}

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

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 $v_{i}$ lies in the one-dimensional space $V_{i - 1}^{⊥} \cap V_{i}$ .