Flag (linear algebra)

In mathematics, particularly in linear algebra, a flag is an increasing sequence of subspaces of a vector space V. Here "increasing" means each is a proper subspace of the next (see filtration):

Related Topics:
Mathematics - Linear algebra - Subspace - Vector space - Filtration

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:{0} = V_0 sub V_1 sub V_2 sub cdots sub V_k = V.

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If we write the dim Vi = di then we have

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:0 = d_0 < d_1 < d_2 < cdots < d_k = n,

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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 di = i, otherwise it is called a partial flag.

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

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The signature of the flag is the sequence d0, d1, … dk.

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