TITLE: Treatment of Linear Equations: Fields vs Polynomials over a field
QUESTION [2 upvotes]: As one starts studying Linear Algebra, the concept of linear equation is presented. Now, in the book I am following, a linear equation of $n$ variables over a field $F$ is said to be the following formula:
$$a_1 x_1 + \dots + a_n x_n = b, \text{where } a_1,\dots,a_n,b \in F.$$
This is crystal clear, however, at least as far as my personal aesthetics is concerned, I would rather treat it over $F[x_1,\dots,x_n]$, such as "something like equalizers" of degree $1$ polynomials over $F$ and degree $0$ polynomials over $F$ (poly of n variables). That is to say, "what I am working with is in my structure", not "flying over the structure". Perhaps it is just misguided aesthetics here, but is there such a treatment out there?
Furthermoe, the nature of $x_i$, while intuitively clear, is understood in naïve and purely intuitive way.
NOTE: I am aware that the cooridnate mapping gives us an isomorphism to $F$, so we might as well stick to the field, but we are clearly working with linear combinations of polynomials here, why do we abandom them in favour of a field? Say I do not fancy coordinates.
Thank you (for shredding some of my confusion)! :)

REPLY [2 votes]: A coordinate free to study linear equations over a vector space $V$ is to use linear functionals. A linear functional is just a linear map $\varphi \colon V \rightarrow \mathbb{F}$ and a linear equation on $V$ is just an expression of the form $\varphi(v) = b$ where $b \in \mathbb{F}$ and $\{ v \in V \, | \, \varphi(v) = b \} = \varphi^{-1}(b)$ is our solution space. More generally, we can consider the solution space of $k$ equations as $\cap_{i=1}^k \varphi^{-1}(b_i)$. The collection of all linear functionals is denoted by $V^{*}$ and called the dual space of $V$. It is itself a vector space (reflecting among other things the fact that we can add equations and multiply them by scalars). Let us see how this works in $\mathbb{F}^n$:
A linear map $\varphi \colon \mathbb{F}^n \rightarrow \mathbb{F}$ can be written uniquely as $\varphi(x_1, \dots, x_n) = a_1 x_1 + \dots + a_n x_n$ for some uniquely determined coefficients $a_i \in \mathbb{F}$. The parameters $x_1, \dots, x_n$ are variables of the function $\varphi$ so we can also think of the elements of the dual space $(\mathbb{F}^n)^{*}$ as linear polynomials in the variables $x_1, \dots, x_n$. A linear equation homogeneous equation has the form
$$ \varphi(x_1, \dots, x_n) = a_1 x_1 + \dots + a_n x_n = 0_{\mathbb{F}} $$
and so the corresponding solution space is $\varphi^{-1}(0) = \ker(\varphi)$. The linear functional $\varphi$ can be multiplied by a scalar $c \in \mathbb{F}$ resulting in a new linear functional $c\varphi$ actings on $(x_1,\dots,x_n)$ by
$$ (c\varphi)(x_1,\dots,x_n) = c x_1 + \dots + c x_n. $$
Thus, we see that $c\varphi = 0$ corresponds to the original equation multiplied by a scalar. Similarly we can add two equations, consider the solution space of many equations (in the case the equations are homogeneous, the space of solutions is called the annahilator of functionals/"equations"), etc.
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