Suppose V is finite-dimensional with dim V ≥ 2.
Prove that there exist S, T ∈ L(V; V ) such that ST ≠ T S.
please assist.
Here's a specific example which holds for V a vector space over any field F. Suppose first that dimV=2; then picking any basis { } for V, we define in that basis the operators as folllows:
Then for any vector we have
but
We see from (3) and (4) and the linear independence of that
unless a=b=0, that is, unless w=0. Thus
as operators in L(V,V). In the event that dimV=n>2, we may build upon the construction of as follows: choosing a basis { } for V, we now define as above, and set
for 3≤i≤n. Then for any we have as above
N1N2(w)≠N2N1(w)
provided at least one of a1,a2≠0. Thus
We have thus shown that for any finite dimensional vector space V over any field F, dimV>1 implies the existence of a noncommutating pair of operators T
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