(from Wikipedia)
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:
$$ \{0\}=V_{0} \subset V_{1} \subset V_{2} \subset \cdots \subset V_{k}=V $$Etymology: suppose I have an actual flag in the sense of a piece of fabric, and I want to explain to someone how to attach it properly to a flagpole. First I need to say which of the four sides should be attached to the flagpole, lest the flag be flown sideways or backwards. Then, on that distinguished side, I need to specify one corner as the top corner, lest the flag be flown upside down. So we have a rectangle with a distinguished edge, and the distinguished edge has a distinguished endpoint.
An ordered basis for _V_ is said to be adapted to a flag _V_0 ⊂ _V_1 ⊂ ... ⊂ 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 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 entry and 0's elsewhere. Concretely, the standard flag is the sequence of subspaces:
$$ 0<\left\langle e_{1}\right\rangle <\left\langle e_{1},e_{2}\right\rangle <\cdots <\left\langle e_{1},\ldots ,e_{n}\right\rangle =K^{n} $$An adapted basis is almost never unique (the counterexamples are trivial).
A complete flag on an [inner product space](https://en.wikipedia.org/wiki/Inner_product_space "Inner product space") has an essentially unique [orthonormal basis](https://en.wikipedia.org/wiki/Orthonormal_basis "Orthonormal basis"): it is unique up to multiplying each vector by a unit (scalar of unit length, e.g. 1, −1, _i_). Such a basis can be constructed using the [Gram-Schmidt process](https://en.wikipedia.org/wiki/Gram-Schmidt_process "Gram-Schmidt process"). The uniqueness up to units follows [inductively](https://en.wikipedia.org/wiki/Mathematical_induction "Mathematical induction"), by noting that $v_i$ lies in the one-dimensional space $V_{i-1}^{\perp }\cap V_{i}$.
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Author of the notes: Antonio J. Pan-Collantes
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