In fact the pathways to advanced mathematics courses all begin with linear algebra and multivariable calculus, and the standard prerequisite for most linear algebra and multivariable calculus ( Real coordinate spaces ) courses includes two semesters of calculus.

because of the tremendous success of analytic geometry (rene descartes), showing that many geometric problems can be solved with algebra, people forget that in a geometry problem, the problem comes first; so, for example, if i have a problem that has a circle and a line not through the circle, i do not start with where the circle is in a pre-defined coordinate plane; instead, since the problem did not specify distances, directions, or sizes, i decide that the given line is my x axis, and the perpendicular to the line through the circle center is my y axis, and i can either choose the circle center to be (0, 1) or i can choose the circle radius to be 1, or whatever is convenient for the problem

the point is that coordinate systems are useful, but they do not define the problem; they come after

yes, according to the level you need to study, I mean at graduate level, postgraduate level, all these courses based on the coordinate system.

it is most convenient to study (and teach) many of the subjects that way, but many parts of math (e.g., topology, category theory, logic) do not have (or need) coordinate systems, and even linear algebra doesn't need coordinate systems for much of its theory (though coordinates are extremely useful for some calculations)

While there was great progress after the development of "analytic geometry" which introduced coordinates, mathematics does not require coordinates. For example in linear algebra we study the category of linear transformations between vector spaces, that is a map T:V->W. Only when both V and W are vector spaces. We can represent T as a matrix. However, the matrix is not "natural" - a it depends on the selection of the basis of V and W and both finite dimensional. An inner or scaler product on a vector space is a bilinear function :V-> reals (or complex) numbers satisfying certain conditions. If one selects a basis one can represent the inter product as a matrix but the representation is not canonical and requires V be finite dimensional, Canonical meaning a In nether case above does the representation as a matrix representation preserve all the structural properties.

That this the representation of a linear map as a matrix (which is dependent on the basis that define the coordinate system) is not an intrinsic property of the map. While the use of a coordinate system can be essential in applications where calculations are necessary, they often are an impediment in the understanding of the intrinsic properties of the object being studied. For example, while any n-dimensional manifold can be embedded into a sufficiently large dimensional Euclidean as a sub manifold - viewing the manifold as a an instinct object not only is more elegant it resulting in better understanding of the geometry and topology. Ditto for viewing algebraic varieties as internist object rather than embedding subvariaties of a projective space. View the mathematics in an intrinsic coordinate free matter often leads to easier proofs (no need to show invariant under coordinate change) and to expressions that are simpler and more related to the geometry.

For example the use of differential forms requires no notion of coordinates and the definition of the differential that makes k forms to (k+1) forms requires no notion of coordinates. Sure one can express them in local coordinates if one wants but they are globally defined independent of coordinates. The definition of a closed k-form w, is simply dw=0 as a (k+1)-form. To look at this in coordinates say in Euclidean three space a closed 1-form is one which the curl of the equivalent vector field vanishes for a 2-from is one whose divergence vanishes. The coordinate based description has no useful information and is more complex - especially when on considering dimensions higher than 3.

exactly as truman prevatt said (more concretely than i did, thanx)

All these problems are based on naive set theory that is good for description of properties of real numbers. Fuzzy set theory is good for description of fuzzy numbers. For the first reading you could to use the old book of Paul Halmos about naive set theory.