\chapter{Why linear relations?}
Linear relations
unmotivated, unintuitive
way of defining linear (in)dependence
Conservation laws (as in physics or accounting) are a key natural
source of linear relations.
\section{Motivation for relations}
"Expressing one vector in terms of the others"
is much more intuitive than
"some combination equals zero"
(and in fact -should- be introduced first)
[In physics, these are refered to as "balance" equations
vs. "conversation" equations:
forces balance out vs. conservation of energy, say;
note that expressing -a single- vector in terms of others
is a rather special kind of balance (perhaps it has a special term?);
you can also think of it as "synthesis".]
This begs the question:
Why bother with linear relations?
Answers:
(there are surprisingly many!)
- canonical
A linear relation does not single out any particular vector
from your set,
and is well-defined up to scale.
(It's a well-defined element of $\cP V^*$.)
- modules
"Express one in terms of others" requires division,
and doesn't generalize to modules.
- subspace
Linear relations form a -subspace- (of $V^*$)
- kernel
(dually, looking at covectors instead of vectors)
A covector, $v^* \in \left(\bR^n\right)^*$,
extends to:
$V^n \to V$
Note that this is -linear-, not multi-linear
(it's just a weighted sum; in terms of morphisms,
a weighted sum of ``project onto the $i$th factor'')
The kernel of this morphism is the sets of vectors
that satisfy this particular linear relation.
\section{Conservation laws}
A key source of linear relations in nature is -conservation laws-.
A + B + C = 0
...and this makes summing to -zero- natural!
(Otherwise it's more intuitive to think of one element
as being a combination of the others.)
In physics,
"conservation of energy, momentum, etc."
and "heat equation" and many, many (linear?) PDEs
are -linear- relations.
In accounting,
the fundamental principle of double-entry accounting is
-conservation of value-
...and "balancing the books" means making sure that all the relations
are satisfied!
Often more intuitively "balance" equations;
e.g. of replicating portfolio in Black-Scholes-Merton model
In calc/infinitesimally,
df = 0
says "you're staying on a level set"
Linear dependence
This illustrates a -key- idea in linear algebra:
rather than stating a -single- equation or set of equations,
you should consider the -space- of all of them.
For instance, the equations
X+Y=0
Y+Z=0
-also- imply X+2Y+Z=0 and 3X+3Y=0 etc.
...indeed, you should take the span in the dual space
(or rather, the span in the dual affine space, but anyways)
to get -all- equations that they satisfy.
...and then one can talk about a -generating- set or -basis-.
[so rank is "how many equations you need to generate all of 'em"
(from the row/equation POV)]
Similarly, if we say that
(1,0), (0,1), (1,1) are -linearly dependent-,
the prosaic way is to say:
"it's redundant: we could do away with (1,1).
Or with (0,1). Or with (1,0)"
Precisely:
(1,1) = (1,0) + (0,1)
...and now you could
% % % % % % % %
Here's a notion that's somewhat natural (and perhaps is thought up by
confused students) but doesn't have a name (that I know of) and isn't
studied:
Given a set of vectors,
if it's linearly dependent,
we can ask if -any proper subset- is linearly dependent,
or whether you need -all- of 'em.
EG,
100, 010, 001, 111
and
100, 010, 001, 110
are both linearly dependent,
but in the 2nd, a 3-element set is linearly dependent.
This is perhaps not asked b/c it's very much about -this specific
set- (not just its span and the # of elements).
Properly, you can of course look at the function
$\cP(X) \to \bN$ given by ``dim of span''
and compare it with ``number of elements''
though this isn't terribly interesting.