Editing Flag (linear algebra) (section)

{{Refimprove|date=September 2014}}
In [[mathematics]], particularly in [[linear algebra]], a '''flag''' is an increasing [[sequence]] of [[Linear subspace|subspaces]] of a [[dimension (vector space)|finite-dimensional]] [[vector space]] ''V''. Here "increasing" means each is a proper subspace of the next (see [[filtration (abstract algebra)|filtration]]):
:<math>\{0\} = V_0 \sub V_1 \sub V_2 \sub \cdots \sub V_k = V.</math>

The term ''flag'' is motivated by a particular example resembling a [[flag]]: the zero point, a line, and a plane correspond to a nail, a staff, and a sheet of fabric.<ref>Kostrikin, Alexei I. and Manin, Yuri I. (1997). ''Linear Algebra and Geometry'', p. 13. Translated from the Russian by M. E. Alferieff. Gordon and Breach Science Publishers. {{ISBN|2-88124-683-4}}.</ref>

If we write that dim''V''<sub>''i''</sub> = ''d''<sub>''i''</sub> then we have
:<math>0 = d_0 < d_1 < d_2 < \cdots < d_k = n,</math>
where ''n'' is the [[dimension (linear algebra)|dimension]] of ''V'' (assumed to be finite). Hence, we must have ''k'' ≤ ''n''. A flag is called a '''complete flag''' if ''d''<sub>''i''</sub> = ''i'' for all ''i'', otherwise it is called a '''partial flag'''.

A partial flag can be obtained from a complete flag by deleting some of the subspaces. Conversely, any partial flag can be completed (in many different ways) by inserting suitable subspaces.

The '''signature''' of the flag is the sequence (''d''<sub>1</sub>, ..., ''d''<sub>''k''</sub>).