\chapter{Linear relations}
Linear relations are a fundamental concept in linear algebra,
but they are generally poorly motivated and explained.
They are usually introduced to give a formal definition of linear independence,
and while elegant, feel opaque and pedantic.
They are actually a natural, intuitive, even familiar concept -- they arise
as conservation laws.
Their algebraic motivation and properties can also be elaborated.
\section{Conservation laws}
Conservation laws are linear relations:
if a quantity is conserved, then
(a change in quantities that make it up sums to zero)
...as in conservation of energy, or accounting equation
[Less cleanly, many can be expressed as balance equations:
dPE = dKE, etc.]
Where do conservation laws arise?
Especially when on some constraint;
if it's an affine space (as in accounting), get global relation;
if it's curved (as in physics?), get infinitesimal relation (on tangent space).
v. frequent in PDEs!!
(heat equation)
\section{Independence of vectors}
What should we mean by independence of vectors, intuitively?
\begin{description}
\item[Non-redundant] You can't express one in terms of others
\item[Unique expression] You can't express a given vector in two different ways
\end{description}
``Unique expression'' generalizes non-redundant:
if $v_1 = \sum_{i \neq 1} a_i v_i$, then these are two different descriptions
for $v_1$ in terms of $v_i$.
Linear relations do encode non-uniqueness of expression,
putting all terms on one side:
it's analogous to conservation equation versus balance equation.
Actually, ``unique expression'' is more general than ``linear relation'',
as it doesn't require subtracting (hence it generalizes not only to modules
over rings, but even over semi-rings? (where you can't necessarily subtract)).
\section{Algebraic}
\subsection{Generalize to modules}
The notion of linear relation generalizes immediately to modules,
and is the correct notion (or at least a good generalization).
For instance, given $2, 3 \in \bZ$,
neither can be expressed as an (integer) multiple of the other,
but they satisfy a linear relation, viz, $3 \cdot 2 + (-2) \cdot 3 = 0$.
In my experience, this is how mathematicians generally make their peace
with the notion, and the story ends there -- at least, this is how it was
for me, and fellow students.
Note that $\set{2,3}$ is a \emph{minimal} spanning set of $\bZ$
in the sense that any proper subset does not span $\bZ$,
but it is not of minimal cardinality.
\subsection{\emph{Space} of linear relations}
express 1 in terms of others / non-unique expression
are intuitive,
but don't have further structure!
Given set of vectors, get the space of all linear relations they satisfy
(subset of $R^k$)
It's kernel of $(a^1,\dots,a^k) \mapsto \sum a^iv_i$
[the matrix whose columns are $v_i$;
this is a map $R^k \to M$]