nLab
linear equation

Idea

Since generally an equation is the statement of equality $\varphi (x)=\psi (y)$ of two functions $\varphi$ and $\psi$ of variables $x$ and $y$, so a linear equation is an equation between linear functions.

In 1-category theory

Typically here a linear function is taken to mean an $R$-linear map over some given ring $R$, hence a homomorphism $\varphi :X\to Z$ or $\psi :Y\to Z$ of $R$-modules $X,Y,Z\in R$Mod. If here $Z$ is an $R$-module of rank greater than 1, one also speaks of a system of linear equations.

Specifically if $R=k$ is a field then these are linear maps of $k$-vector spaces and hence in this case a linear equation is a statement of equality of two vectors $\varphi (x)=\psi (y)$ in some vector space $Z$ that depend linearly on vectors $x$ in a vector spaces $X$ and $y\in Y$.

Frequently this is considered specifically for the case that $g$ is a constant function, hence just a fixed vector. In this case the linear equation becomes $\varphi (x)=g$ for $x\in X$. If moreover $\varphi$ here is represented or representable by a matrix this is typically written as

$$\varphi \cdot \stackrel{\rightharpoonup }{x}=\stackrel{\rightharpoonup }{g},$$\phi \cdot\vec x = \vec g \,,

which is the form that one finds in standard textbooks on linear algebra. If $\stackrel{\rightharpoonup }{g}=0$ here this is called a homogeneous linear equation.

But linear equations make sense and are important in the greater generality where $R$ is not necessarily a field, and in fact in contexts more general than that of $R$-modules even. For instance natural isomorphisms between linear functors are a kind of categorification of linear equations.

In $(\mathrm{\infty },1)$-category theory

Indeed, as discussed at equation, in the formal logic of type theory a an equation as above is a judgement of the form

$$x:X,y:Y⊢(\varphi (x)=\psi (y)):\mathrm{Type}$$x : X , y : Y \vdash (\phi(x) = \psi(y)) : Type

whose solution space is the dependent sum

$$\mathrm{Sol}≔⊢\sum _{\genfrac{}{}{0ex}{}{x:X}{y:Y}}(\varphi (x)=\psi (y)):\mathrm{Type}$$Sol \;\; \coloneqq \vdash \sum_{{x : X} \atop {y : Y}} (\phi(x) = \psi(y)) : Type

and reading this in fact in homotopy type theory says that $X,Y,\mathrm{Sol}$ are homotopy types.

The generalization of a ring $R$ to a homotopy type is an E-∞-ring and that of an $R$-module $X,Y$ is a module spectrum.

Accordingly a linear equation in homotopy(type) homotopy theory is a statement of equivalence between elements of an $R$-module spectrum that depend $R$-linearly on other $R$-module spectra. More precisely, as discussed at equation, the solution space to such an ”$\mathrm{\infty }$-linear equation” is the homotopy pullback

$$\begin{array}{ccc}X{\times }_Z^{\mathrm{\infty }}Y& \to & Y\\ \downarrow & ⇙& {\downarrow }^{\psi }\\ X& \underset{\varphi }{\to }& Z\end{array}$$\array{ X \times_Z^\infty Y &\to& Y \\ \downarrow &\swArrow& \downarrow^{\mathrlap{\psi}} \\ X &\underset{\phi}{\to}& Z }

in an (∞,1)-category of $R$-∞-modules.

Properties

Solution spaces of homogeneous $R$-linear equations

We discuss solution space of homogeneous linear equations in the general context of linear algebra over a ring $R$ (not necessarily a field).

So let $R$ be a ring and let $N\in R$Mod be an $R$-module.

Let $n_0,n_1\in \mathbb{N}$ and let $K=(K_{ij})$ be an an $n_0\times n_1$ matrix with entries in $R$. By matrix multiplication this defines a linear function

$$K\cdot (-):N^{n_0}\to N^{n_1}.$$K \cdot (-) : N^{n_0} \to N^{n_1} \,.

which takes the element $\stackrel{\rightharpoonup }{u}=(u_1,\cdots ,u_{n_0})\in N^{n_0}$ to the element $K\cdot \stackrel{\rightharpoonup }{u}$ with

$$(K\cdot \stackrel{\rightharpoonup }{u}{)}_i=\sum _{j=1}^{n_0}{K}_{ij}\cdot u_j.$$(K \cdot \vec u)_i = \sum_{j = 1}^{n_0} K_{i j}\cdot u_j \,.

Consider dually the linear map

$$(-)\cdot K^T:R^{n_1}\to R^{n_0}$$(-) \cdot K^T : R^{n_1} \to R^{n_0}

on the free modules over $R$.

Consider the quotient module of $R^{n_1}$ by the image of this map

$$R^{n_1}/(R^{n_0}\cdot K^T),$$R^{n_1}/ (R^{n_0} \cdot K^T) \,,

hence the cokernel of the map, fitting in the exact sequence

$$R^{n_1}\stackrel{(-)\cdot K}{\to }{R}^{n_0}\to R^{n_1}/(R^{n_1}\cdot K^T)\to 0$$R^{n_1} \stackrel{(-)\cdot K}{\to} R^{n_0} \to R^{n_1}/(R^{n_1}\cdot K^T) \to 0

Here the morphism on the left is also called the inclusion of the syzygies of the module on the right.

Applying the left exact hom functor ${\mathrm{Hom}}_{R\mathrm{Mod}}(-,N)$ to this yields exact sequence

$$0\to {\mathrm{Hom}}_{R\mathrm{Mod}}(R^{n_1}/(R^{n_0}\cdot K^T),N)\to N^{n_0}\stackrel{K\cdot (-)}{\to }{N}^{n_1}.$$0 \to Hom_{R Mod}(R^{n_1}/(R^{n_0}\cdot K^T), N) \to N^{n_0} \stackrel{K \cdot(-)}{\to} N^{n_1} \,.

This identifies ${\mathrm{Hom}}_{R\mathrm{Mod}}(R^{n_1}/(R^{n_0}\cdot K),N)$ as the space of solutions of the homogeneous linear equation $K\cdot \stackrel{\rightharpoonup }{u}=0$.

(…)

Relation to syzygies and projective resolutions of modules

For $R$ a ring, there is close relation between

1. $R$-linear equations in finitely many variables;

2. finitely generated $R$-modules;

3. syzygies in these $R$-modules

4. and projective resolutions of these $R$-modules.

These relations we discuss in the following.

(…)

Related concepts

• differential equation

• matrix calculus

• determinant

References

Discussion in the context of syzygies and projective resolutions of modules is for instance in section 4.5 of

• Pierre Schapira, Categories and homological algebra, lecture notes (2011) (pdf)