One striking feature of Noether's 1929 paper occurred in a footnote: “As B. L. van der Waerden has communicated to me, one can obtain an invariant connection, independent of the specific choice of basis, by separating the concepts linear transformation and matrix. A linear transformation is a homomorphism of two modules of linear forms; a matrix is an expression (the representation) of this homomorphism by a definite choice of basis” [76,670]. Here we have the essential modern connection between the notions of linear transformation, matrix, and module (or vector space). Two years later, van der Waerden's insight about the proper way of viewing that connection was presented to a much wider audience in his textbook.
Moore, Gregory H. “The Axiomatization of Linear Algebra: 1875-1940.” Historia Mathematica, volume 22, issue 3, 1995, pp. 262–303. ScienceDirect, doi:10.1006/hmat.1995.1025.