Flags in A Vector Space
A flag in a finite dimensional vector space V over a field F is an increasing sequence of subspaces, where "increasing" means each is a proper subspace of the next (see filtration):
If we write the dim Vi = di then we have
where n is the dimension of V. Hence, we must have k ≤ n. A flag is called a complete flag if di = i, otherwise it is called a partial flag. The signature of the flag is the sequence (d1, … dk).
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.
Read more about this topic: Generalized Flag Variety
Famous quotes containing the words flags in, flags and/or space:
“No doubt I shall go on writing, stumbling across tundras of unmeaning, planting words like bloody flags in my wake. Loose ends, things unrelated, shifts, nightmare journeys, cities arrived at and left, meetings, desertions, betrayals, all manner of unions, adulteries, triumphs, defeats ... these are the facts.”
—Alexander Trocchi (1925–1983)
“Still, it is dear defiance now to carry
Fair flags of you above my indignation,”
—Gwendolyn Brooks (b. 1917)
“In bourgeois society, the French and the industrial revolution transformed the authorization of political space. The political revolution put an end to the formalized hierarchy of the ancien regimé.... Concurrently, the industrial revolution subverted the social hierarchy upon which the old political space was based. It transformed the experience of society from one of vertical hierarchy to one of horizontal class stratification.”
—Donald M. Lowe, U.S. historian, educator. History of Bourgeois Perception, ch. 4, University of Chicago Press (1982)