## Bases and Flags

An ordered basis for a vector space\[V\]

is said to be adapted to a flag if the first \[d_i\]

basis vectors form a basis for each \[0 \leq i \leq k\]

. 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. The standard flag in \[\mathbb{R}^n\]

is induced from the standard basis \[\left\{ \mathbf{e}_1 , ..., \mathbf{e}_n \right\}\]

. The standard flag is the sequence of subspaces:\[0 \lt \left\{ \mathbf{e}_1 \right\} \lt \left\{ \mathbf{e}_1 , \mathbf{e}_2 \right\} \lt ⋯ \lt \left\{ \mathbf{e}_1 , … , \mathbf{e}_n \right\} = \mathbb{R}^n\]

}An adapted basis is almost never unique.

A complete flag on an inner product space has an essentially unique orthonormal basis up to multiplying each vector by a constant of size 1.