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)
“The flags are natures newly found.
Rifles grow sharper on the sight.
There is a rumble of autumnal marching,
From which no soft sleeve relieves us.
Fate is the present desperado.”
—Wallace Stevens (1879–1955)
“What a phenomenon it has been—science fiction, space fiction—exploding out of nowhere, unexpectedly of course, as always happens when the human mind is being forced to expand; this time starwards, galaxy-wise, and who knows where next.”
—Doris Lessing (b. 1919)