Editing Flag (linear algebra) (section)

==Bases==
An ordered [[basis (linear algebra)|basis]] for ''V'' is said to be '''adapted''' to a flag ''V''<sub>0</sub> ⊂ ''V''<sub>1</sub> ⊂ ... ⊂ ''V''<sub>''k''</sub> if the first ''d''<sub>''i''</sub> basis vectors form a basis for ''V''<sub>''i''</sub> 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''<sub>''i''</sub> be the [[linear span|span]] of the first ''i'' basis vectors. For example, the '''{{Visible anchor|standard flag}}''' in '''R'''<sup>''n''</sup> is induced from the [[standard basis]] (''e''<sub>1</sub>, ..., ''e''<sub>''n''</sub>) where ''e''<sub>''i''</sub> denotes the vector with a 1 in the ''i''th entry and 0's elsewhere. Concretely, the standard flag is the sequence of subspaces:
:<math>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.</math>

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 [[mathematical induction|inductively]], by noting that <math>v_i</math> lies in the one-dimensional space <math>V_{i-1}^\perp \cap V_i</math>.

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]].<ref>Harris, Joe (1991). ''Representation Theory: A First Course'', p. 95. Springer. {{ISBN|0387974954}}.</ref>