nLab linear equation




Since generally an equation is the statement of equality ϕ(x)=ψ(y)\phi(x) = \psi(y) of two functions ϕ\phi and ψ\psi of variables xx and yy, so a linear equation is an equation between linear functions.

In 1-category theory

Typically here a linear function is taken to mean an RR-linear map over some given ring RR, hence a homomorphism ϕ:XZ\phi : X \to Z or ψ:YZ\psi : Y \to Z of (say left) RR-modules X,Y,ZRX, Y, Z \in RMod. If here ZZ is an RR-module of rank greater than 1, one also speaks of a system of linear equations.

Specifically if R=kR = k is a field then these are linear maps of kk-vector spaces and hence in this case a linear equation is a statement of equality of two vectors ϕ(x)=ψ(y)\phi(x) = \psi(y) in some vector space ZZ that depend linearly on vectors xx in a vector spaces XX and yYy \in Y.

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

ϕx=g, \phi \cdot\vec x = \vec g \,,

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

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

In (,1)(\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(ϕ(x)=ψ(y)):Type x : X , y : Y \vdash (\phi(x) = \psi(y)) : Type

whose solution space is the dependent sum

Sol x:Xy:Y(ϕ(x)=ψ(y)):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,SolX, Y, Sol are homotopy types.

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

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

X× Z Y Y ψ X ϕ Z \array{ X \times_Z^\infty Y &\to& Y \\ \downarrow &\swArrow& \downarrow^{\mathrlap{\psi}} \\ X &\underset{\phi}{\to}& Z }

in an (∞,1)-category of RR-∞-modules.


Solution spaces of homogeneous RR-linear equations

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

So let RR be a ring and let NRN \in RMod be an RR-module.

Let n 0,n 1n_0,n_1 \in \mathbb{N} and let K=(K ij)K = (K_{i j}) be an an n 0×n 1n_0 \times n_1 matrix with entries in RR. By matrix multiplication this defines a linear function

K():N n 0N n 1. K \cdot (-) : N^{n_0} \to N^{n_1} \,.

which takes the element u=(u 1,,u n 0)N n 0\vec u = (u_1, \cdots, u_{n_0}) \in N^{n_0} to the element KuK \cdot \vec u with

(Ku) i= j=1 n 0K iju j. (K \cdot \vec u)_i = \sum_{j = 1}^{n_0} K_{i j}\cdot u_j \,.

Consider dually the linear map

()K T:R n 1R n 0 (-) \cdot K^T : R^{n_1} \to R^{n_0}

on the free modules over RR.

Consider the quotient module of R n 1R^{n_1} by the image of this map

R n 1/(R n 0K T), R^{n_1}/ (R^{n_0} \cdot K^T) \,,

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

R n 1()KR n 0R n 1/(R n 1K T)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 Hom RMod(,N)Hom_{R Mod}(-,N) to this yields exact sequence

0Hom RMod(R n 1/(R n 0K T),N)N n 0K()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 Hom RMod(R n 1/(R n 0K),N)Hom_{R Mod}(R^{n_1}/(R^{n_0}\cdot K), N) as the space of solutions of the homogeneous linear equation Ku=0K \cdot \vec u = 0.


Relation to syzygies and projective resolutions of modules

For RR a ring, there is close relation between

  1. RR-linear equations in finitely many variables;

  2. finitely generatedRR-modules;

  3. syzygies in these RR-modules

  4. and projective resolutions of these RR-modules.

These relations we discuss in the following.



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

Linear equations over skewfields were studied in

  • Oystein Ore, Linear equations in non-commutative fields, Ann. Math. 32:3 (1931) 463–477 doi

and more lately in the theory of quasideterminants, see there.

category: algebra

Last revised on May 21, 2024 at 09:39:20. See the history of this page for a list of all contributions to it.