Suppose V is finite-dimensional with dim V greater or equal to 2. Prove that there exist S, T €L(V, V) such that ST is not equal to T S
solution:-
N1,N2 L(V,V) as follows
N1(1) =0, N1(2 )=1 ; (1)
N2(1)=2 ,N2(2)=0 (2)
Then for any vector w=a1 + b2 we have
N2N1(w)= aN2N1(1)+bN2N1(2)=b2 (3)
but
N1N2(w)=aN1N2(1) + bN1N2(2)=a1 (4)
we see from 3 and 4 the linear independence of 1,2 that
N1N2(w) N2N1(w)
unless a=b=0 that is ,unless w=0. Thus,
N1N2 N2N1
as operators in L(V,V). In the event that dim V=n>2, we may build upon a construction of N1,N2follows : choosing a basis {1,2......,n } for V, we define N1,N2 on 1,2
as the above set
N1(i)=N2(i)=0
for 3in, then for any w= aiiV we have as above
N1N2(w) N2N1(w)
providing at least one of a1,a2 0, thus
N1N2 N2N1.
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