Starting with an abstract formal description of vector spaces and linear transformations is not a pedagogically useful introduction, and it leaves out a huge am

Starting with an abstract formal description of vector spaces and linear transformations is not a pedagogically useful introduction, and it leaves out a huge amount of the historical/motivational context explaining most of the conventions used in linear algebra, and even the mental/conceptual understanding most working mathematicians have about the meaning of linear models and their use in various parts of mathematics (not even to mention people in other STEM fields). The two places to start historically (and easily accessible to high school students) are: (1) understanding and working with displacement vectors in 3-dimensional Euclidean affine space and in general thinking about transformation geometry (sometime later this can be extended to other kinds of non-Euclidean or non-metrical geometry), especially with reference to problems in Newtonian mechanics. (2) systems of linear equations: this one is inherently coordinate-heavy and matrix based, at least to start out, and explains our conventions for how matrices are written, index order, multiplication of matrices by "column" vectors on the right, the use of matrix equations to fold several equals signs into one, etc. After that I'd call out the manipulation of vectors of polynomial coefficients as an accessible additional concrete example of a linear space. Discussion of other linear spaces or more purely abstract treatments proving properties from axioms can come sometime later, after students have familiarity with some of those tools, and after they have applied them to some problems in statistics, multivariable calculus, computer graphics, ODEs, optimization, etc.