## 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.