## Nested Sequence of Vector Spaces and Flags

In linear algebra, a flag is a strictly nesting sequence of subspaces of a finite dimensional vector space\[V\]

.\[\mathbf{0} \subset V_1 \subset V_2 \subset ... \subset V_n =V\]

If the dimension of

\[V_i\]

is \[d_i\]

then \[d_0 \lt d_1 \lt d_2 \lt d_3 \lt ...\lt d_k =n\]

where

\[n\]

is the dimension of \[V\]

. A flag is called a complete flag if \[d_i =i\]

, otherwise it is called a partial flag. The sequence \[d_1, \: d_2 , \: d_3, \: d_4 \: ... \: d_k\]

is called the signature of the flag.