Question #209440

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


1
Expert's answer
2021-06-24T12:37:31-0400

solution:-

N1,N2 \in L(V,V) as follows

N1(υ\upsilon1) =0, N1(υ\upsilon2 )=υ\upsilon(1)

N2(υ\upsilon1)=υ\upsilon,N2(υ\upsilon2)=0 (2)


Then for any vector w=aυ\upsilon1 + bυ\upsilon2 we have


N2N1(w)= aN2N1(υ\upsilon1)+bN2N1(υ\upsilon2)=bυ\upsilon(3)

but


N1N2(w)=aN1N2(υ\upsilon1) + bN1N2(υ\upsilon2)=aυ\upsilon1 (4)



we see from 3 and 4 the linear independence of υ\upsilon1,υ\upsilon2 that

N1N2(w)\neq N2N1(w)

unless a=b=0 that is ,unless w=0. Thus,

N1N2 \neq 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 {υ\upsilon1,υ\upsilon2......,υ\upsilonn } for V, we define N1,N2 on υ\upsilon1,υ\upsilon2


as the above set

N1(υ\upsiloni)=N2(υ\upsiloni)=0

for 3\leqi\leqn, then for any w=\sum aiυ\upsiloni\inV we have as above

N1N2(w) \neq N2N1(w)

providing at least one of a1,a2\neq 0, thus

N1N2\neq N2N1.




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