Suppose that
a
ij
= estimated effect on patient
i
of drug
j
x
j
= quantity of drug
j
in the mixture
then the effect of drug mixture on patient
i
is
a
i
1
x
1
+
a
i
2
x
2
+
· · ·
+
a
in
x
n
UCSD Center for Computational Mathematics
Slide 5/40, Wednesday, January 12th, 2011
Problem: If
b
i
is the
desired effect
of drug mixture on patient
i
,
find
x
1
,
x
2
, . . . ,
x
n
, such that
b
i
=
a
i
1
x
1
+
a
i
2
x
2
+
· · ·
+
a
in
x
n
for
i
= 1, 2, . . . ,
m
This is the same as: find
x
1
,
x
2
, . . . ,
x
n
, such that
b
i
=
n
X
j
=1
a
ij
x
j
for
i
= 1, 2, . . . ,
m
UCSD Center for Computational Mathematics
Slide 6/40, Wednesday, January 12th, 2011
In matrix form:
b
=
Ax
, where
b
=
b
1
b
2
.
.
.
b
m
,
A
=
a
11
a
12
· · ·
a
1
n
a
21
a
22
. . .
a
2
n
.
.
.
.
.
.
.
.
.
.
.
.
a
m
1
a
m
1
· · ·
a
mn
,
x
=
x
1
x
2
.
.
.
x
n
This is a
system of linear equations
.
UCSD Center for Computational Mathematics
Slide 7/40, Wednesday, January 12th, 2011
Another notation: write
A
by columns:
A
=
a
1
a
2
· · ·
a
n

{z
}
columns of
A
with
a
j
=
a
1
j
a
2
j
.
.
.
a
mj
In this case,
a
j
= effects of drug
j
with
a
j
∈
R
m
, i.e.,
Ax
=
n
X
j
=1
a
j
x
j
UCSD Center for Computational Mathematics
Slide 8/40, Wednesday, January 12th, 2011