Suppose S1,…,SnS_1,\dots,S_nS1,…,Sn are injective linear maps such that S1S2…SnS_1S_2\dots S_nS1S2…Sn makes sense. Prove that S1S2…SnS_1S_2\dots S_nS1S2…Sn is injective.
Obvious, as none of them can take a nonzero vector to zero applying all of them in sequence still can't take a nonzero vector to zero.