Bases and Flags

An ordered basis for a vector space  
  is said to be adapted to a flag if the first  
  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  
  be the span of the first  
  basis vectors. The standard flag in  
  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.

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