Biology Reference
In-Depth Information
The basis
P
found above is an example of “base-changing” from a mathematically
valid basis
of
N
(
S
) to one that is mathematically
and
biochemically valid. Question
to ask at this juncture include:
Is it always possible to find a mathematically valid basis for the nullspace
N
(
S
)ofa
stoichiometric matrix
S
? If so, is it always possible to find a biochemically valid basis
P
B
of the nullspace of a stoichiometric
matrix
S
? The first we can answer; the second is a research question which we will
illustrate in another example.
Changing bases of a vector space can also be viewed in terms of “changing
coordinates.” A set
, starting from a mathematically valid basis
B
of basis vectors for an (arbitrary) vector space
W
can be viewed as a set of “coordinates” for
W
. This literally means that any vector
w
B
={
z
1
,...,
z
t
}
∈
W
has a unique expression as a linear combination in terms of the elements of
B
. More precisely, there are real numbers
w
1
,...,w
t
so that
w
=
w
1
z
1
+···+
w
t
z
t
,
and if
w
∈
W
is any other vector, then
w
=
w
if and only if, in the corresponding
expression
w
=
w
1
z
1
+···+
w
t
z
t
,
w
1
=
w
1
,...,w
t
=
w
t
.
One can represent
w
one has
∈
W
by listing the coefficients
w
j
of
w
in the linear combi-
nation to give
w
=
w
1
z
1
+···+
w
t
z
t
T
. In this way, every vector
w
in
W
corresponds uniquely
=
(w
1
,...,w
t
)
as a vector
w
to a “point”
(w
1
,...,w
t
)
.
Exercise 8.14.
2
, the usual Euclidean plane. Let
e
1
be the standard unit vector
(vector of length one) in the direction of the positive
x
-axis, and let
e
2
be the
standard unit vector in the direction of the positive
y
-axis. Use geometric prop-
erties of vector operations in Euclidean space to sketch the vector
a.
Suppose
W
=
R
(
−
4
)
e
1
+
5
e
2
.
To what ordered pair does this linear combination of vectors correspond?
b.
However, now replace
e
1
by
b
1
:=
2
be the
2
e
1
+
3
e
2
, and let
b
2
=
e
2
.Let
w
∈
R
vector whose coordinates are (
−
4, 5) in the new coordinates given by
{
b
1
,
b
2
}
.
Sketch
w
using the old coordinate system
{
e
1
,
e
2
}
.
Going back to metabolic pathways and stoichiometric matrices, complete the fol-
lowing exercise.
Exercise 8.15.
a.
Define
and the vector
x
as in Exercise
8.13
(f).
Find a vector in the corresponding system
N
(
S
) which has two distinct represen-
tations as linear combinations of elements of
A
to consist of the elements of
P
. (This reflects a general principle:
One cannot add a new vector to a set that is already a basis and still have a basis.
Be able to justify this!)
A
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