Biology Reference
In-Depth Information
b.
Express each vector of
(coming as before from (
8.14
)) as an element of
N
(
S
),
with coordinates coming from the basis
B
P
.
,
for the nullspace
N
(
S
) of the stoichiometric matrix
S
as in Exercises
8.5
and
8.8
.As
per Exercises
8.13
and
8.15
,dim
Exercises
8.13
and
8.15
explore two bases, namely the sets of vectors
B
and
P
6, so any vector in
N
(
S
) can be expressed
uniquely in six coordinates taken either with respect to
(
N
(
S
))
=
B
, or with respect to
P
.For
11
, we previously checked
example, viewing
N
(
S
) as a subspace of Euclidean space
R
that
T
).
(As per usual conventions, the coordinates here are in terms of the standard basis
{
v
:=
(
1
,
1
,
0
,
0
,
0
,
1
,
0
,
−
1
,
0
,
0
,
1
)
∈
N
(
S
11
.) However, in terms of coordinates with respect to the ordered
e
1
,...,
e
11
}
of
R
basis
B
={
z
1
,...,
z
6
}
of
N
(
S
) as a six-dimensional vector space,
T
v
=
z
1
=
(
1
,
0
,
0
,
0
,
0
,
0
)
B
.
(8.15)
Likewise, since
y
1
=
z
1
, in terms of coordinates with respect to the ordered set
P
={
y
1
,...,
y
6
}
, one has
T
v
=
(
1
,
0
,
0
,
0
,
0
,
0
)
P
.
(8.16)
On the other hand,
T
11
w
:=
(
0
,
−
1
,
1
,
0
,
0
,
−
1
,
1
,
0
,
0
,
0
,
0
)
∈
N
(
S
)
⊂
R
,
satisfies
T
w
=
z
4
=
(
0
,
0
,
0
,
1
,
0
,
0
)
B
,
(8.17)
but
T
=
(
−
,
,
,
,
,
)
P
=
y
4
−
y
1
.
w
1
0
0
1
0
0
(8.18)
A
change-of-basis matrix
9
from the basis
B
to the basis
P
is a square 6
×
6matrix
A
with the following property:
B,P
If
T
u
:=
(
u
1
,...,
u
6
)
B
,
:=
then the matrix product
q
A
B,P
u
gives the coordinate expression for
u
in the basis
P
, i.e.,
T
q
=
(
q
1
,...,
q
6
)
P
.
The matrix
A
can be created by setting each of its columns
[
A
]
∗
j
,
1
j
6, to
B
,
P
be the basis vector
z
j
∈
B
written in terms of coordinates of
P
. Thus, in this case, by
Eqs. (
8.15
) and (
8.16
),
T
[
A
]
∗
1
=
(
1
,
0
,
0
,
0
,
0
,
0
)
,
while by Eqs. (
8.17
) and (
8.18
),
T
[
A
]
∗
4
=
(
−
1
,
0
,
0
,
1
,
0
,
0
)
.
9
Also called a “transition matrix” in some texts, while in others, this term is reserved for Markov chain
processes.
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