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Let $V$ be a finite-dimensional vector space. A filtration of subspaces $$V_1\subset V_2\subset\cdots \subset V_n= V$$ is called a flag in $V$ . We speak of a complete flag when $$\dim V_i = i$$ for each $i=1,\ldots,n$ .
Next, putting $$d_k = \dim V_k,\quad k=1,\ldots n,$$ we say that a list of vectors $(u_1,\ldots,u_{d_n})$ is an adapted basis relative to the flag, if the first $d_1$ vectors give a basis of $V_1$ , the first $d_2$ vectors give a basis of $V_2$ , etc. Thus, an alternate characterization of a complete flag, is that the first $k$ elements of an adapted basis are a basis of $V_k$
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Let us consider $\reals^n$ . For each $k=1,\ldots,n$ let $V_k$ be the span of $e_1,\ldots,e_k$ , where $e_j$ denotes the $j\supth$ basic vector, i.e. the column vector with $1$ in the $j\supth$ position and zeros everywhere else. The $V_k$ give a complete flag in $\reals^n$ . The list $(e_1,e_2,\ldots, e_n)$ is an adapted basis relative to this flag, but the list $(e_2,e_1,\ldots,e_n)$ is not.
More generally, a flag can be defined as a maximal chain in a partially ordered set. If one considers the poset consisting of subspaces of a (finite dimensional) vector space, one recovers the definition given above.
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"flag" is owned by rmilson. [ full author list (3) ]
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| Also defines: |
adapted basis, complete flag |
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Cross-references: finite dimensional, partially ordered set, chain, column vector, span, elements, characterization, basis, vectors, subspaces, filtration, vector space, finite-dimensional
There are 8 references to this entry.
This is version 6 of flag, born on 2002-06-01, modified 2006-09-23.
Object id is 2994, canonical name is Flag.
Accessed 6299 times total.
Classification:
| AMS MSC: | 15A03 (Linear and multilinear algebra; matrix theory :: Vector spaces, linear dependence, rank) | | | 06A06 (Order, lattices, ordered algebraic structures :: Ordered sets :: Partial order, general) |
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Pending Errata and Addenda
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