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\begin{document}

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\section*{linear equation}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{equality_and_equivalence}{}\paragraph*{{Equality and Equivalence}}\label{equality_and_equivalence}

[[!include equality and equivalence - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{in_1category_theory}{In 1-category theory}\dotfill \pageref*{in_1category_theory} \linebreak
\noindent\hyperlink{in_category_theory}{In $(\infty,1)$-category theory}\dotfill \pageref*{in_category_theory} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{solution_spaces_of_homogeneous_linear_equations}{Solution spaces of homogeneous $R$-linear equations}\dotfill \pageref*{solution_spaces_of_homogeneous_linear_equations} \linebreak
\noindent\hyperlink{relation_to_syzygies_and_projective_resolutions_of_modules}{Relation to syzygies and projective resolutions of modules}\dotfill \pageref*{relation_to_syzygies_and_projective_resolutions_of_modules} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

Since generally an \emph{[[equation]]} is the statement of [[equality]] $\phi(x) = \psi(y)$ of two [[functions]] $\phi$ and $\psi$ of [[variables]] $x$ and $y$, so a \emph{linear equation} is an equation between [[linear functions]].

\hypertarget{in_1category_theory}{}\subsubsection*{{In 1-category theory}}\label{in_1category_theory}

Typically here a \emph{[[linear function]]} is taken to mean an \emph{$R$-[[linear map]]} over some given [[ring]] $R$, hence a [[homomorphism]] $\phi : X \to Z$ 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 \emph{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]] $\phi(x) = \psi(y)$ 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 $\phi(x) = g$ for $x \in X$. If moreover $\phi$ here is represented or representable by a [[matrix]] this is typically written as

\begin{displaymath}
\phi \cdot\vec x = \vec g
  \,,
\end{displaymath}
which is the form that one finds in standard textbooks on [[linear algebra]]. If $\vec g = 0$ here this is called a \emph{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.

\hypertarget{in_category_theory}{}\subsubsection*{{In $(\infty,1)$-category theory}}\label{in_category_theory}

Indeed, as discussed at \emph{[[equation]]}, in the [[formal logic]] of [[type theory]] a an equation as above is a [[judgement]] of the form

\begin{displaymath}
x : X , y : Y \vdash (\phi(x) = \psi(y)) : Type
\end{displaymath}
whose solution space is the [[dependent sum]]

\begin{displaymath}
Sol \;\; \coloneqq \vdash \sum_{{x : X} \atop {y : Y}} (\phi(x) = \psi(y)) : Type
\end{displaymath}
and reading this in fact in [[homotopy type theory]] says that $X, Y, Sol$ are \emph{[[homotopy types]]}.

The generalization of a [[ring]] $R$ to a [[homotopy type]] is an \emph{[[E-∞-ring]]} and that of an $R$-[[module]] $X, Y$ is a \emph{[[module spectrum]]}.

Accordingly a linear equation in [[homotopy]]([[homotopy type theory|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 \emph{[[equation]]}, the solution space to such an ``$\infty$-linear equation'' is the [[homotopy pullback]]

\begin{displaymath}
\itexarray{
    X \times_Z^\infty Y &\to& Y
    \\
    \downarrow &\swArrow& \downarrow^{\mathrlap{\psi}}
    \\
    X &\underset{\phi}{\to}& Z
  }
\end{displaymath}
in an [[(∞,1)-category]] of $R$-[[∞-modules]].

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{solution_spaces_of_homogeneous_linear_equations}{}\subsubsection*{{Solution spaces of homogeneous $R$-linear equations}}\label{solution_spaces_of_homogeneous_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 $K = (K_{i j})$ be an an $n_0 \times n_1$ [[matrix]] with entries in $R$. By [[matrix multiplication]] this defines a [[linear function]]

\begin{displaymath}
K \cdot (-) : N^{n_0} \to N^{n_1}
  \,.
\end{displaymath}
which takes the element $\vec u = (u_1, \cdots, u_{n_0}) \in N^{n_0}$ to the element $K \cdot \vec u$ with

\begin{displaymath}
(K \cdot \vec u)_i = \sum_{j = 1}^{n_0} K_{i j}\cdot u_j
  \,.
\end{displaymath}
Consider dually the linear map

\begin{displaymath}
(-) \cdot K^T : R^{n_1} \to R^{n_0}
\end{displaymath}
on the [[free modules]] over $R$.

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

\begin{displaymath}
R^{n_1}/ (R^{n_0} \cdot K^T)
  \,,
\end{displaymath}
hence the [[cokernel]] of the map, fitting in the [[exact sequence]]

\begin{displaymath}
R^{n_1} \stackrel{(-)\cdot K}{\to} R^{n_0} \to R^{n_1}/(R^{n_1}\cdot K^T)
  \to 0
\end{displaymath}
Here the morphism on the left is also called the inclusion of the [[syzygies]] of the module on the right.

Applying the [[left exact functor|left exact]] [[hom functor]] $Hom_{R Mod}(-,N)$ to this yields [[exact sequence]]

\begin{displaymath}
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}
  \,.
\end{displaymath}
This identifies $Hom_{R Mod}(R^{n_1}/(R^{n_0}\cdot K), N)$ as the space of solutions of the \emph{homogeneous} linear [[equation]] $K \cdot \vec u = 0$.

(\ldots{})

\hypertarget{relation_to_syzygies_and_projective_resolutions_of_modules}{}\subsubsection*{{Relation to syzygies and projective resolutions of modules}}\label{relation_to_syzygies_and_projective_resolutions_of_modules}

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

\begin{enumerate}%
\item $R$-linear equations in finitely many variables;


\item [[generators and relations|finitely generated]] $R$-[[modules]];


\item [[syzygies]] in these $R$-modules


\item and [[projective resolutions]] of these $R$-modules.



\end{enumerate}
These relations we discuss in the following.

(\ldots{})

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[differential equation]]


\item [[matrix calculus]]


\item [[determinant]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

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

\begin{itemize}%
\item [[Pierre Schapira]], \emph{Categories and homological algebra}, lecture notes (2011) (\href{http://people.math.jussieu.fr/~schapira/lectnotes/HomAl.pdf}{pdf})

\end{itemize}
[[!redirects linear equations]]



\end{document}