Linear algebra, one of the core courses in modern mathematics, has previously been asserted to have developed in "the West." Its main problem is the solution of systems of n linear equations in n unknowns. Previous accounts have attributed the origins of its two main techniques to preeminent European mathematicians, namely, Leibniz (1646-1716), for his work on determinants, and Gauss (1777-1855), for his work on elimination. Thus, although it is well known that the earliest record of what we now call Gaussian elimination is found in the Nine Chapters on the Mathematical Arts (Jiuzhang suanshu, c. 1st century CE), it has been assumed that linear algebra developed independently in the West, without any connection to developments in China. This paper presents evidence of the diffusion of linear algebra across the Eurasian continent, from early imperial China to medieval Europe. More specifically, linear algebra problems with determinantal-style solutions so distinctive that they can serve as "fingerprints" are recorded in Chinese mathematical treatises dating from the first century CE and in Fibonacci's treatises dating from the thirteenth century. More broadly, this evidence suggests the need to rethink the world history of science prior to the scientific revolution. That is, that problems this specialized, with solutions this esoteric, eventually spread across Eurasia -- including early imperial China and thirteenth-century Italy -- suggests that the assumption that other mathematical and scientific practices were not similarly transmitted should be reconsidered. To do so, we must reconsider the relationship between scientific practices, texts, and authorship during this period. Scientific practices of this period often did not depend on texts: the learning, teaching, and transmission of these practices did not require literacy; when these practices were recorded in texts, it was commonly for purposes of patronage or displays of expertise. Instead, extant texts preserve only fragmentary evidence of practices of the period. It thus makes little sense to assign credit to what we now anachronistically call "China" or "the West" based merely on the earliest known extant text in which a practice is recorded. It is likely that scientific practices circulated following routes by which commerce, art, and religion were transported by traders, missionaries, and travelers, during what is increasingly understood to be the "global Middle Ages."