| Flag (linear Algebra) |
Article Index for Flag |
Website Links For Flag |
Information AboutFlag (linear Algebra) |
|
: If we write the dim ''V''''i'' = ''d''''i'' then we have : where ''n'' is the Dimension of ''V'' (assumed to be finite-dimensional). Hence, we must have ''k'' ≤ ''n''. A flag is called a complete flag if ''d''''i'' = ''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''0, ''d''1, … ''d''''k''. BASES An ordered Basis for ''V'' is said to be adapted to a flag if the first ''d''''i'' basis vectors form a basis for ''V''''i'' 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''''i'' be the span of the first ''i'' basis vectors. For example, the standard flag in R''n'' is induced from the Standard Basis {''e''1, …, ''e''''n''} where ''e''''i'' denotes the vector with a 1 in the ''i''th slot and 0's elsewhere. SUBSPACE NEST In an infinite-dimensional space ''V'', as used in Functional Analysis , the flag idea generalises to a subspace nest, namely a collection of subspaces of ''V'' that is a Total Order for inclusion and which further is closed under arbitrary intersections and closed linear spans. See Nest Algebra . SEE ALSO |
|
|