# Do (systems of linear equations with scalars and unknowns from different algebraic structures) occur widely?

Generally in linear algebra one studies systems of linear equations where both coefficients and unknowns belong to the same field. I would not be the first person to notice that a system like $$\begin{array}{rrrcl}\mathbf{x}_1&+\mathbf{x}_2&+2\mathbf{x}_3&=&\mathbf{b}_1\\\mathbf{x}_1&+\mathbf{x}_2&+\mathbf{x}_3&=&\mathbf{b}_2\end{array}$$ (scalars in $\mathbb{R}$ and unknowns & 'constants' $\mathbf{x}_i,\mathbf{b}_j \in V$ where $V$ is an arbitrary vector space over $\mathbb{R}$) can be solved by row-reduction, assignment of free parameters and back-substitution to get $$\left[\begin{array}{c}\mathbf{x}_1\\\mathbf{x}_2\\\mathbf{x}_3\end{array}\right] = \left[\begin{array}{c}2\mathbf{b}_1-\mathbf{b}_1\\\mathbf{0}_V\\\mathbf{b}_1-\mathbf{b}_2\end{array}\right] + \left[\begin{array}{c}-\mathbf{t}\\\mathbf{t}\\\mathbf{0}_V\end{array}\right] = \left[\begin{array}{c}2\mathbf{b}_1-\mathbf{b}_1\\\mathbf{0}_V\\\mathbf{b}_1-\mathbf{b}_2\end{array}\right] + \left[\begin{array}{c}-1\\1\\0_{\mathbb{R}}\end{array}\right] \left[\begin{array}{c}\mathbf{t}\end{array}\right] \text{ for } \mathbf{t} \in V$$ where $\left[\mathbf{t}\right]$ is a $1\times1$ matrix. Parameters do not factor out the same way as when $V=\mathbb{R}$.

Wikipedia mentions linear equations over a ring and over polynomials (Gröbner basis theory), but in both cases coefficients and unknowns are from the same algebraic structure.

My question is: are there broad areas in mathematics where such systems of equations (scalars from a field, unknowns from a general vector space) occur? Or, more generally, are there broad classes of linear equations with scalars and unknowns from different algebraic structures?

Yes. Here is a sketch of an example.

Suppose that you want to understand the behavior of the solutions a system of equations as the "parameters" of the equations vary.

To be concrete, imagine we have the equation:

$a_0 x_0 + a_1 x_1 = 0$

Here we can think of the $a_i$ as $\mathbb{R}$-valued functions on some space $X$, which you can take as a first approximation to be the line $\mathbb{R}$. We will take our functions $a_i$ to be polynomials in $\mathbb{R}[t]$.

So at each point $p$ we have a fixed equation $a_0(p) x_0 + a_1(p) x_1 = 0$, and these equations vary nicely if we move around on $X$ nicely (nice can be: continuous, smooth, algebraic, etc.).

Above every point on $X$ we can put a copy of $\mathbb{R}^2$, with coordinates $x_i$, and so above each point on $X$ there is a corresponding set of solutions to the equations. These solutions "move" in a algebraic fashion as we move around on $X$.

Maybe at this point you should try to draw a picture of this. Over each point in the line there should be a line of solutions (or a plane possibly if $a_0 = a_1 = 0$), and your polynomial functions $a_i$ control how it moves.

I said above that taking $X$ to be the line $\mathbb{R}$ is only a first approximation. In fact, one gets a more flexible theory if you include other points in this line - this is a mysterious point to explain without going into technical details, so please just accept it for now.

For example, we will associate a point to the entire line $P$, or a point to the pair $(i,-i)$. (These correspond in some fashion to the polynomials $(0)$ and $(t^2 + 1)$, respectively.)

Once this is done, it no longer makes sense to think of the $a_i$ as being real valued functions everywhere - over the point $(i,-i)$ they will take complex values, and over the point $P$ they will take values in the field of rational functions $\mathbb{R}(t)$. But note that since we chose polynomials in $\mathbb{R}[t]$ for the $a_i$, they still know how to be functions on $\mathbb{C}$, or $\mathbb{R}(t)$: $t$ evaluated at $(i,-i)$, which amounts to the map $R[t] \to R[t]/(t^2 + 1) \cong \mathbb{C}$ is $i$, $t$ evaluated at $P$ (the map $R[t] \to R(t)$) is $t$.

And over these points it makes sense for the same reasons to put the vector spaces $\mathbb{C}^2$ or $\mathbb{R}(t)^2$ over the corresponding points (after all, the equations now have complex coefficients, so it makes less sense to look for solutions $\mathbb{R}^2$). In each of these vector spaces, the functions $a_i$ define an equation $a_ix_i =0$, which defines a subspace.

Maybe you can try to draw a (rough) picture now? (I certainly can't, but it helps to try.) One should still get the sense that the set of solutions moves nicely, even though the space in which it lives may change from point to point.

So the moral of this story is that there are situations where you can study equations whose fields of coefficients change in some controlled way. As you saw, the situation is described by something vaguely geometric.

The study of these objects is called scheme theory, which is a topic in algebraic geometry.