▁if ▁there ▁exists ▁a ▁basis ▁such ▁that ▁every ▁element ▁of ▁ Σ ▁has ▁an ▁upper - tri angular ▁matrix ▁representation ▁in ▁that ▁basis . ▁It ▁follows ▁from ▁Burn side ' s ▁theorem ▁that ▁every ▁commut ative ▁algebra ▁ Σ ▁in ▁L ( V ) ▁is ▁tri angular izable . ▁Hence ▁every ▁comm uting ▁family ▁in ▁L ( V ) ▁can ▁be ▁simultaneously ▁upper - tri angular ized . ▁▁ ▁Left ▁ide als ▁▁ ▁If ▁A ▁is ▁an ▁algebra , ▁one ▁can ▁define ▁a ▁left ▁regular ▁representation ▁ Φ ▁on ▁A : ▁ Φ ( a ) b ▁= ▁ab ▁is ▁a ▁hom omorphism ▁from ▁A ▁to ▁L ( A ), ▁the ▁algebra ▁of ▁linear ▁transformations ▁on ▁A ▁ ▁The ▁invariant ▁sub spaces ▁of ▁ Φ ▁are ▁precisely ▁the ▁left ▁ide als ▁of ▁A . ▁A ▁left ▁ideal ▁M ▁of ▁A ▁gives ▁a ▁sub re presentation ▁of ▁A ▁on ▁M . ▁ ▁If ▁M ▁is ▁a ▁left ▁ideal ▁of ▁A . ▁Consider ▁the ▁quot ient ▁vector ▁space ▁A / M . ▁The ▁left ▁regular ▁representation ▁ Φ ▁on ▁M ▁now ▁desc ends ▁to ▁a ▁representation ▁ Φ ' ▁on ▁A / M . ▁If ▁[ b ] ▁denotes ▁an ▁equivalence ▁class ▁in ▁A / M , ▁ Φ '( a )[ b ] ▁= ▁[ ab ]. ▁The ▁kernel ▁of ▁the ▁representation ▁ Φ ' ▁is ▁the ▁set ▁{ a ▁ ∈ ▁A ▁| ▁ab ▁ ∈ ▁M ▁for ▁all ▁b }. ▁ ▁The ▁representation ▁ Φ ' ▁is ▁ir reducible ▁if ▁and ▁only