On second thoughts, I expressed myself very poorly... it goes without saying that matrices and linear algebra are identical; what I meant was that one can compute quantum interactions with the S-Matrix alone and not need to ‘explicitly’ use linear algebra concepts to deal with superposition of states & cetra. The focus on the s-matrix as a fundamental formalism was spearheaded by Wheeler, Heisenberg, and Landau (and amongst others), if my memory serves.

P.S. Even this statement is formally quite weak, but hopefully I have clarified myself enough to transfer my intended meaning. Apologies.