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In this paper, we present Ginkgo, a modern C++ math library for scientific high performance computing. While classical linear algebra libraries act on matrix and vector objects, Ginkgo's design principle abstracts all functionality as "linear operators", motivating the notation of a "linear operator algebra library". Ginkgo's current focus is oriented towards providing sparse linear algebra functionality for high performance GPU architectures, but given the library design, this focus can be easily extended to accommodate other algorithms and hardware architectures. We introduce this sophisticated software architecture that separates core algorithms from architecture-specific back ends and provide details on extensibility and sustainability measures. We also demonstrate Ginkgo's usability by providing examples on how to use its functionality inside the MFEM and deal.ii finite element ecosystems. Finally, we offer a practical demonstration of Ginkgo's high performance on state-of-the-art GPU architectures.