Let be a field. For any , let . Prove that is a minimal left ideal in .
I know how to prove that is a left ideal in , but having trouble with the minimality part of the proof. Can I get some help please?
let be a left ideal of which is contained in for some i'll prove that let where and is an element of which has 1 in the position and 0 elsewhere. so there exists some such that now let . then, since is a left ideal of , we have so for all
and hence because is a basis for