▁upon ▁the ▁underlying ▁scalar ▁field ▁of ▁V . ▁ ▁An ▁invariant ▁vector ▁( fixed ▁point ▁of ▁T ), ▁other ▁than ▁ 0 , ▁sp ans ▁an ▁invariant ▁sub space ▁of ▁dimension ▁ 1 . ▁An ▁invariant ▁sub space ▁of ▁dimension ▁ 1 ▁will ▁be ▁acted ▁on ▁by ▁T ▁by ▁a ▁scalar ▁and ▁consists ▁of ▁invariant ▁vectors ▁if ▁and ▁only ▁if ▁that ▁scalar ▁is ▁ 1 . ▁ ▁As ▁the ▁above ▁examples ▁indicate , ▁the ▁invariant ▁sub spaces ▁of ▁a ▁given ▁linear ▁transformation ▁T ▁shed ▁light ▁on ▁the ▁structure ▁of ▁T . ▁When ▁V ▁is ▁a ▁finite - dimensional ▁vector ▁space ▁over ▁an ▁algebra ically ▁closed ▁field , ▁linear ▁transformations ▁acting ▁on ▁V ▁are ▁character ized ▁( up ▁to ▁similarity ) ▁by ▁the ▁Jordan ▁canonical ▁form , ▁which ▁decom poses ▁V ▁into ▁invariant ▁sub spaces ▁of ▁T . ▁Many ▁fundamental ▁questions ▁regarding ▁T ▁can ▁be ▁translated ▁to ▁questions ▁about ▁invariant ▁sub spaces ▁of ▁T . ▁ ▁More ▁generally , ▁invariant ▁sub spaces ▁are ▁defined ▁for ▁sets ▁of ▁operators ▁as ▁sub spaces ▁invariant ▁for ▁each ▁operator ▁in ▁the ▁set . ▁Let ▁L ( V ) ▁denote ▁the ▁algebra ▁of ▁linear ▁transformations ▁on ▁V , ▁and ▁Lat ( T ) ▁be ▁the ▁family ▁of ▁sub spaces ▁invariant ▁under ▁T ▁ ∈ ▁L ( V ). ▁( The ▁" Lat " ▁notation ▁refers ▁to ▁the ▁fact ▁that ▁Lat ( T ) ▁forms ▁a ▁lattice ; ▁see ▁discussion ▁below .) ▁Given ▁a ▁non empty ▁set ▁ Σ ▁ ⊂ ▁L ( V ), ▁one ▁cons iders ▁the ▁invariant