▁sub spaces ▁invariant ▁under ▁each ▁T ▁ ∈ ▁ Σ . ▁In ▁symbols , ▁ ▁For ▁instance , ▁it ▁is ▁clear ▁that ▁if ▁ Σ ▁= ▁L ( V ), ▁then ▁ ▁Lat ( Σ ) ▁= ▁{ ▁{ 0 }, ▁V }. ▁ ▁Given ▁a ▁representation ▁of ▁a ▁group ▁G ▁on ▁a ▁vector ▁space ▁V , ▁we ▁have ▁a ▁linear ▁transformation ▁T ( g ) ▁: ▁V ▁→ ▁V ▁for ▁every ▁element ▁g ▁of ▁G . ▁If ▁a ▁sub space ▁W ▁of ▁V ▁is ▁invariant ▁with ▁respect ▁to ▁all ▁these ▁transformations , ▁then ▁it ▁is ▁a ▁sub re presentation ▁and ▁the ▁group ▁G ▁acts ▁on ▁W ▁in ▁a ▁natural ▁way . ▁ ▁As ▁another ▁example , ▁let ▁T ▁ ∈ ▁L ( V ) ▁and ▁ Σ ▁be ▁the ▁algebra ▁generated ▁by ▁{ 1 ,   T }, ▁where ▁ 1 ▁is ▁the ▁identity ▁operator . ▁Then ▁Lat ( T ) ▁= ▁Lat ( Σ ). ▁Because ▁T ▁lies ▁in ▁ Σ ▁trivial ly , ▁Lat ( Σ ) ▁ ⊂ ▁Lat ( T ). ▁On ▁the ▁other ▁hand , ▁ Σ ▁consists ▁of ▁polynomials ▁in ▁ 1 ▁and ▁T , ▁therefore ▁the ▁reverse ▁inclusion ▁holds ▁as ▁well . ▁ ▁Matrix ▁representation ▁▁ ▁Over ▁a ▁finite ▁dimensional ▁vector ▁space ▁every ▁linear ▁transformation ▁T ▁: ▁V ▁→ ▁V ▁can ▁be ▁represented ▁by ▁a ▁matrix ▁once ▁a ▁basis ▁of ▁V ▁has ▁been ▁chosen . ▁ ▁Suppose ▁now ▁W ▁is ▁a ▁T ▁invariant ▁sub space . ▁Pick ▁a ▁basis ▁C ▁= ▁{ v 1 ,