▁linear ▁mapping ▁ ▁from ▁some ▁vector ▁space ▁V ▁to ▁itself ▁is ▁a ▁sub space ▁W ▁of ▁V ▁such ▁that ▁T ( W ) ▁is ▁contained ▁in ▁W . ▁An ▁invariant ▁sub space ▁of ▁T ▁is ▁also ▁said ▁to ▁be ▁ ▁T ▁invariant . ▁ ▁If ▁W ▁is ▁T - in variant , ▁we ▁can ▁restrict ▁T ▁to ▁W ▁to ▁arrive ▁at ▁a ▁new ▁linear ▁mapping ▁▁▁▁ ▁Next , ▁we ▁give ▁a ▁few ▁immediate ▁examples ▁of ▁invariant ▁sub spaces . ▁ ▁C ertain ly ▁V ▁itself , ▁and ▁the ▁sub space ▁{ 0 }, ▁are ▁trivial ly ▁invariant ▁sub spaces ▁for ▁every ▁linear ▁operator ▁T ▁: ▁V ▁→ ▁V . ▁For ▁certain ▁linear ▁operators ▁there ▁is ▁no ▁non - t rivial ▁invariant ▁sub space ; ▁consider ▁for ▁instance ▁a ▁rotation ▁of ▁a ▁two - dimensional ▁real ▁vector ▁space . ▁ ▁Let ▁v ▁be ▁an ▁eigen vector ▁of ▁T , ▁i . e . ▁T ▁v ▁= ▁ λ v . ▁Then ▁W ▁= ▁span ▁{ v } ▁is ▁T ▁invariant . ▁As ▁a ▁consequence ▁of ▁the ▁fundamental ▁theorem ▁of ▁algebra , ▁every ▁linear ▁operator ▁on ▁a ▁complex ▁finite - dimensional ▁vector ▁space ▁with ▁dimension ▁at ▁least ▁ 2 ▁has ▁an ▁eigen vector . ▁Therefore , ▁every ▁such ▁linear ▁operator ▁has ▁a ▁non - t rivial ▁invariant ▁sub space . ▁The ▁fact ▁that ▁the ▁complex ▁numbers ▁are ▁algebra ically ▁closed ▁is ▁required ▁here . ▁Compar ing ▁with ▁the ▁previous ▁example , ▁one ▁can ▁see ▁that ▁the ▁invariant ▁sub spaces ▁of ▁a ▁linear ▁transformation ▁are ▁dependent