8.13
Affine combinations of solutions of linear equations.
Consider the set of
m
linear equations
in
n
variables
Ax
=
b
, where
A
is an
m
×
n
matrix,
b
is an
m
vector, and
x
is the
n
vector
of variables. Suppose that the
n
vectors
z
1
, . . . , z
k
are solutions of this set of equations,
i.e.
, satisfy
Az
i
=
b
. Show that if the coefficients
α
1
, . . . , α
k
satisfy
α
1
+
· · ·
+
α
k
= 1,
then the affine combination
w
=
α
1
z
1
+
· · ·
+
α
k
z
k
is a solution of the linear equations,
i.e.
, satisfies
Aw
=
b
. In words: Any affine combina
tion of solutions of a set of linear equations is also a solution of the equations.
8.14
Stoichiometry and equilibrium reaction rates.
We consider a system (such as a single cell)
containing
m
metabolites (chemical species), with
n
reactions among the metabolites
occurring at rates given by the
n
vector
r
. (A negative reaction rate means the reaction
runs in reverse.) Each reaction consumes some metabolites and produces others, in known
rates proportional to the reaction rate. This is specified in the
m
×
n
stoichiometry matrix
S
, where
S
ij
is the rate of metabolite
i
production by reaction
j
, running at rate one.
(When
S
ij
is negative, it means that when reaction
j
runs at rate one, metabolite
i
is
consumed.) The system is said to be in equilibrium if the total production rate of each
metabolite, due to all the reactions, is zero.
This means that for each metabolite, the
total production rate balances the total consumption rate, so the total quantities of the
metabolites in the system do not change.
Express the condition that the system is in
equilibrium as a set of linear equations in the reaction rates.
162
8
Linear equations
Chapter 9
Linear dynamical systems
In this chapter we consider a useful application of matrixvector multiplication,
which is used to describe many systems or phenomena that change or evolve over
time.
9.1
Linear dynamical systems
Suppose
x
1
, x
2
, . . .
is a sequence of
n
vectors. The index (subscript) denotes time
or period, and is written as
t
;
x
t
, the value of the sequence at time (or period)
t
, is called the
state
at time
t
. We can think of
x
t
as a vector that changes over
time,
i.e.
, one that changes dynamically. In this context, the sequence
x
1
, x
2
, . . .
is
sometimes called a
trajectory
or
state trajectory
. We sometimes refer to
x
t
as the
current state
of the system (implicitly assuming the current time is
t
), and
x
t
+1
as
the
next state
,
x
t

1
as the
previous state
, and so on.
The state
x
t
can represent a portfolio that changes daily, or the positions and
velocities of the parts of a mechanical system, or the quarterly activity of an econ
omy. If
x
t
represents a portfolio that changes daily, (
x
5
)
3
is the amount of asset 3
held in the portfolio on (trading) day 5.
A
linear dynamical system
is a simple model for the sequence, in which each
x
t
+1
is a linear function of
x
t
:
x
t
+1
=
A
t
x
t
,
t
= 1
,
2
,
. . . .
(9.1)
Here the
n
×
n
matrices
A
t
are called the
dynamics matrices
. The equation above
is called the
dynamics
or
update
equation, since it gives us the next value of
x
,
i.e.
,
x
t
+1
, as a function of the current value
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 Fall '17
 The American, The Land