## 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.