Biology Reference
In-Depth Information
Exercise 8.16.
a.
Complete the remaining four columns of the change-of-basis matrix
A
,from
B,P
basis
B
to
P
, using the results of Exercise
8.13
:
⎡
⎣
⎤
⎦
1
−
1
0
0
0
0
A
B
,
P
=
.
0
1
0
0
0
0
T
B
b.
Use
A
.
Also
, what vector of
N
(
S
) does
u
represent, expressed in terms of the standard
coordinates in
to find the coordinates of
u
:=
(
1
,
3
,
0
,
−
1
,
2
,
1
)
in terms of
P
B
,
P
11
?
R
T
B
c.
Use
A
tofind the coordinates of an arbitrary vector
c
:=
(
c
1
,
c
2
,
c
3
,
c
4
,
c
5
,
c
6
)
B
,
P
in terms of
P
.
In the next exercise, you will repeat the previous exercise, but switch the roles of
B
.
Exercise 8.17.
and
P
a.
Using the results of Exercise
8.15
, now find the change-of-basis matrix
A
P,B
from basis
P
to basis
B
of
N
(
S
).
T
P
b.
Use
A
.
Also
, what vector of
N
(
S
) does
v
represent, expressed in terms of the standard
coordinates in
to find the coordinates of
v
:=
(
1
,
3
,
0
,
−
1
,
2
,
1
)
in terms of
B
P,B
11
?
R
c.
Use
A
to find the coordinates of an arbitrary vector
d
:=
(
d
1
,
d
2
,
d
3
,
d
4
,
P,B
T
P
.
d.
Compute the matrix products
A
B,P
A
P,B
and
A
P,B
A
B,P
, and explain your
answer.
d
5
,
d
6
)
in terms of
B
The “Rank + Nullity Theorem”
10
says that, for an arbitrary matrix
Exercise 8.18.
A
∈
M
m
,
n
(
R
)
,
dim
(
N
(
A
))
+
rank
(
A
)
=
n
,
where the dimension dim
of the nullspace of
A
is often called the
nullity
of
A
, and rank (
A
), the rank of
A
, is the dimension of the row space of
A
, equivalently,
the number of nonzero rows in
E
A
.
Suppose for parts (a) - (d) and (f),
S
(
N
(
A
))
∈
M
m
,
n
(
R
)
is a generic stoichiometric matrix:
a.
Recall:
What is the biological meaning of the number
n
of columns of
S
?
b.
Recall:
What is the biological meaning of the number
m
of rows of
S
?
10
This is sometimes just called the “Rank Theorem.”
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