Question

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.

Answer 1

Let $v_1,v_2,...,v_n$ be a basis of $V$.

Define $S,T\in L(V,V)$ such that

$S(v_l)= \begin{cases} v_2 & \text{if }l=1 \\ 0 & otherwise \end{cases} \\ T(v_l)= \begin{cases} v_1 & \text{if }l=2 \\ 0 & otherwise \end{cases}$

Consider

$(ST)(v_1)=S(T(v_1))=S(0)=0$, since $S$ is linear.

But

$(TS)(v_1)=T(S(v_1))=T(v_2)=v_1 \neq 0$

So $ST \neq TS$.