I know a bit of abstract algebra and of things like group theory. However, I know little beyond the very basic examples of the field. Forgive me for the lack of rigor or precise definition in what I describe, for I am only describing things as I see them, without understanding the proper language or definitions.
I've been studying differential forms and vector spaces of linear functions, but am having trouble understanding rigorously certain details. Let me give a concrete example of what may concern me. Take a Jacobian of form:
$$ \left( \begin{array}{cc} {\partial f \over \partial x}(a) & {\partial f \over \partial y}(a) \\ {\partial f \over \partial x}(b) & {\partial f \over \partial x}(b) \\ \end{array} \right) = \left( \begin{array}{cc} {\partial f \over \partial x} & {\partial f \over \partial y}\\ {\partial f \over \partial x} & {\partial f \over \partial x} \\ \end{array} \right) \cdot \left( \begin{array}{cc} {a} \\ {b} \\ \end{array} \right) $$ I am most confused on this specific array: $$\left( \begin{array}{cc} {\partial f \over \partial x} & {\partial f \over \partial y}\\ {\partial f \over \partial x} & {\partial f \over \partial x} \\ \end{array} \right) $$ It appears as if these functions are almost "floating" without being defined on any domain (not even the real number line or some $x$. Although sin($a$) and cos($a$) are not linear operators, it seems a bit to me like writing "multiplication" of the form:
$$ \mathrm{sin}(a) + \mathrm{cos}(a) = (\mathrm{sin} + \mathrm{cos})(a) $$ Where sin and cos seem to be "floating" without any map to a domain, and $a$ is some real number. If you continue to write "multiplication" out you could get something like: $$ (b) (\mathrm{sin} + \mathrm{cos}) (a) = b \cdot\mathrm{sin}(a) + b \cdot\mathrm{cos}(a) \neq a \cdot\mathrm{sin}(b) + a \cdot\mathrm{cos}(b) = (a) (\mathrm{sin} + \mathrm{cos}) (b)$$ In this case if you "multiply" from the left side, these "floating" sin and cos values behave like vectors, and the left-handed multiplication behaves like scalar multiplication of a vector. But if you "multiply" from the right side, you are performing the mapping of the function to some real-numbered value. I'm wondering what these "floating" functions are called professionally, and what this algebraic structure is. I believe that the left-handed multiplication does in fact satisfy a vector space, but this definition of right-handed multiplication seems to add a bit of extra structure, and defines "something" else. Any recommendations for further reading would be appreciated, and thank you in advance.