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. Concretely, the standard flag is the subspaces:
An adapted basis is almost never unique (trivial counterexamples); 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, like 1, -1, i). This is easiest to prove inductively, by noting that, which defines it uniquely up to unit.
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.
Read more about this topic: Flag (linear Algebra)
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