>>10901057I think the earlier you can separate vectors/linear transformations and coordinates, the better off you'll be. Matrix multiplication, addition, and the determinant are all coordinate operations which can be seen as arising from coordinate-free geometric concepts.
In real vector spaces, vectors can essentially be thought of as "arrows" emanating from the origin, such that you can add them and scale them in the usual way.
UPON CHOOSING A BASIS, you can associate each vector with a tuple of coordinates uniquely.
Given two spaces V and W, a linear map f:V-->W is simply a function which "preserves lines", i.e. commutes with vector addition and rescaling. UPON CHOOSING A BASIS, a linear map is determined by its behavior on the basis vectors, and so can be represented using coordinates. The matrix representation is useful, as (a) addition of linear maps as functions corresponds to coordinatewise addition of the matrices, and (b) the composite of two linear maps is represented by the matrix product of their matrices.
The determinant is easier to define using the coordinates, but can also be understood geometrically. There is something called an "exterior power" of a vector space which has to do with length, area, volume, ... measurements on the space. The nth exterior power of a n-dimensional space is always one-dimensional, and hence will be isomorphic to the underlying scalar field, thought of as a space over itself. For R-spaces, this will always be R and can be thought of as the space of "signed volumes" you can measure in V. This construction is functorial, so any linear map f:V-->V can be turned into a map f*:R-->R, such that if some region in V has volume r, then it is mapped to a region with volume f*(r). The canonical basis for R is the number 1, here representing unit volume, so that f* can be thought of as multiplication by a single number, det(f). This number det(f) tells you how much the volume of the unit cube is rescaled by f.
