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

The term flag is motivated by a particular example resembling a flag: the zero point, a line, and a plane correspond to a nail, a staff, and a sheet of fabric.

If we write that dimV_i = d_i then we have

where n is the dimension of V (assumed to be finite). Hence, we must have k ≤ n. A flag is called a complete flag if d_i = i for all 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 (d₁, ..., d_k).

An ordered basis for V is said to be adapted to a flag V₀ ⊂ V₁ ⊂ ... ⊂ V_k 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 Rⁿ is induced from the standard basis (e₁, ..., e_n) where e_i denotes the vector with a 1 in the ith entry and 0's elsewhere. Concretely, the standard flag is the sequence of subspaces: