Exercise 3.B.11

Proof. We will give a proof by induction.

Clearly, the hypothesis is true when $n=1$. Assume it is true for $n$.

Let $v$ and $v^{\prime }$ be two vectors in the domain of $S_{n+1}$ such that $S_1{S}_2\dots S_{n+1}v=S_1{S}_2\dots S_{n+1}v$. By induction, it follows that $S_1{S}_2\dots S_n$ is injective and, hence, $S_{n+1}v=S_{n+1}{v}^{\prime }$. Since $S_{n+1}$ is also injective, we have that $v=v^{\prime }$, completing the proof. □