Problem
Suppose $V$ is finite-dimensional with $\dim V \geq 2$. Prove that there exist $S,T \in \mathcal{L}(V,V)$ such that $ST \neq TS$
Solution
Let $v_1, v_2$ be a linearly independent list of $V$ which exist as per the restriction $\dim V \geq 2$ and definition 2.36. Now we can uniquely the linear maps $S$ and $T$ as the extension described in P3A10 to the subspace $U = \textrm{span}(v_1, v_2)$ and where $T,S$ is defined uniquely on $U$ by theorem 3.5 and the following: $$Sv_1 = v_2 \thinspace\thinspace\thinspace\thinspace\thinspace\thinspace Sv_2 = v_1$$ $$Tv_1 = v_1 \thinspace\thinspace\thinspace\thinspace\thinspace\thinspace Tv_2 = -v_2$$ Then it follows that: $$STv_2 = S(-v_2) \stackrel{D3.2}{=} -Sv_2 = -v_1$$ $$TSv_2 = Tv_1 = v_1$$ But as $v_1 \neq 0$ by the fact that $v_1$ is a part of a linearly independent list implying $ST \neq TS$.