sub space ▁lattice ▁of ▁ Σ ▁and ▁denoted ▁by ▁Lat ( Σ ). ▁ ▁The ▁lattice ▁operations ▁are ▁defined ▁in ▁a ▁natural ▁way : ▁for ▁ Σ ' ▁ ⊂ ▁ Σ , ▁the ▁meet ▁operation ▁is ▁defined ▁by ▁ ▁while ▁the ▁join ▁operation ▁is ▁ ▁A ▁minimal ▁element ▁in ▁Lat ( Σ ) ▁in ▁said ▁to ▁be ▁a ▁minimal ▁invariant ▁sub space . ▁▁ ▁Fund amental ▁theorem ▁of ▁non comm ut ative ▁algebra ▁▁ ▁Just ▁as ▁the ▁fundamental ▁theorem ▁of ▁algebra ▁ens ures ▁that ▁every ▁linear ▁transformation ▁acting ▁on ▁a ▁finite ▁dimensional ▁complex ▁vector ▁space ▁has ▁a ▁n ont rivial ▁invariant ▁sub space , ▁the ▁fundamental ▁theorem ▁of ▁non comm ut ative ▁algebra ▁as ser ts ▁that ▁Lat ( Σ ) ▁contains ▁n ont rivial ▁elements ▁for ▁certain ▁ Σ . Theorem ▁( B urn side ) ▁Assume ▁V ▁is ▁a ▁complex ▁vector ▁space ▁of ▁finite ▁dimension . ▁For ▁every ▁proper ▁sub algebra ▁ Σ ▁of ▁L ( V ), ▁Lat ( Σ ) ▁contains ▁a ▁n ont rivial ▁element . ▁ ▁Burn side ' s ▁theorem ▁is ▁of ▁fundamental ▁importance ▁in ▁linear ▁algebra . ▁One ▁consequence ▁is ▁that ▁every ▁comm uting ▁family ▁in ▁L ( V ) ▁can ▁be ▁simultaneously ▁upper - tri angular ized . ▁ ▁A ▁non empty ▁ Σ ▁ ⊂ ▁L ( V ) ▁is ▁said ▁to ▁be ▁tri angular izable ▁if ▁there ▁exists ▁a ▁basis ▁{ e 1 ... en } ▁of ▁V ▁such ▁that ▁ ▁In ▁other ▁words , ▁ Σ ▁is ▁tri angular izable