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It is well-known that a quiver Q of type A_n is representation-finite, andthat its indecomposable representations are thin (all Jordan-Hoeldermultiplicities are 0 or 1). By now, various methods of proof are known. The aimof this note is to provide a straight-forward arrangement of possible argumentsin order to avoid indices and clumsy inductive considerations, but alsoavoiding somewhat fancy tools such as the Bernstein-Gelfand-Ponomarevreflection functors or bilinear forms and root systems. The proof we presentdeals with representations in general, not only finite-dimensional ones. Weonly use first year linear algebra, namely the existence of bases of vectorspaces V, W compatible with a given linear map V -> W, and the existence of abasis of a vector space which is compatible with two given subspaces.

The representations of quivers of type A_n. A fast approach