Let $T : V \to W$ be a linear transformation between finite-dimensional vector spaces. Given ordered bases $B_V = { v_1, \dots, v_n }$ of $V$ and $B_W = { w_1, \dots, w_m }$ of $W$, the matrix representation $[T]_{B_W}^{B_V}$ satisfies
$T(v_k) = a_{1k} w_1 + a_{2k} w_2 + \cdots + a_{mk} w_m $,
so that the $k$-th column of the matrix consists of the coordinates of $T(v_k)$ with respect to $B_W$.
Now, when we think of a system of linear equations
$A\mathbf{x} = \mathbf{b}$,
we can interpret the matrix $A$ as defining a linear transformation
$T : \mathbb{R}^n \to \mathbb{R}^m, \quad T(\mathbf{x}) = A\mathbf{x}$.
In this case, each row of $A$ corresponds to a linear equation — that is, to a linear functional on $\mathbb{R}^n$.
I understand that, in the coordinate view, the codomain basis of $\mathbb{R}^m$ is the standard one
$B_W = \{ e_1, e_2, \dots, e_m \}$,
where $e_i$ has a $1$ in the $i$-th position and zeros elsewhere.
However, I’m interested in the functional (non-coordinate) view — analogous to how, in the space of polynomials $P_2$, we can describe the basis ${1, x, x^2}$ corresponding to evaluation of coefficients.
