Page 56 - Matrices theory and applications
P. 56
(c) Show that E 1 ,... ,E n are in direct sum. Deduce that each E j
is a line.
(d) Show that there exist generators e j of each E j such that M jk e l =
l
δ e j .
k
(e) Deduce that every algebra automorphism of M n (k)is interior:
For every σ ∈ AutM n(k), there exists P ∈ GL n (k) such that
σ(M)= P −1 MP For every M ∈ M n(k). 2.9. Exercises 39
