Lecture 5
Review of Linear Equations
UCSD Math 171A: Numerical Optimization
Philip E. Gill
(
[email protected]
)
Wednesday, January 14th, 2009
Review of linear equations
We review properties of systems of linear equations
Ax
=
b
where
A
is an
m
×
n
matrix
and
b
is an
m
vector.
We say
A
∈
R
m
×
n
and
b
∈
R
m
.
We make no assumptions on the
shape of A
⇒
we cannot say that
x
=
A

1
b
, in general.
UCSD Center for Computational Mathematics
Slide 2/35, Wednesday, January 14th, 2009
Example:
finding the drug mixtures in a chemotherapy treatment
Set of
m
typical patient types:
patient 1
,
patient 2
,
. . . ,
patient
m
Set of
n
chemotherapy drugs:
drug 1
,
drug 2
,
. . . ,
drug
n
x
j
= quantity of drug
j
in the medication
a
ij
= estimated effect on patient
i
of drug
j
b
i
= desired effect of drug mixture on patient
i
UCSD Center for Computational Mathematics
Slide 3/35, Wednesday, January 14th, 2009
Problem:
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:
b
i
=
n
X
j
=1
a
ij
x
j
for
i
= 1, 2, . . . ,
m
UCSD Center for Computational Mathematics
Slide 4/35, Wednesday, January 14th, 2009
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In matrix form:
b
=
Ax
, where
A
=
a
11
a
12
· · ·
a
1
n
a
21
a
22
. . .
a
2
n
.
.
.
.
.
.
.
.
.
.
.
.
a
m
1
a
m
1
· · ·
a
mn
,
b
=
b
1
b
2
.
.
.
b
m
,
x
=
x
1
x
2
.
.
.
x
n
This is a system of linear equations.
UCSD Center for Computational Mathematics
Slide 5/35, Wednesday, January 14th, 2009
Another notation.
Write
A
by columns:
A
=
a
1
a
2
· · ·
a
n
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 6/35, Wednesday, January 14th, 2009
An obvious question:
Is it possible to get the desired effect using this mixture?
i.e., does there exist an
x
such that
b
=
Ax
?
UCSD Center for Computational Mathematics
Slide 7/35, Wednesday, January 14th, 2009
Definition
A system of equations is
compatible
if there exists a vector
x
such
that
b
=
Ax
.
i.e., there exist
x
1
,
x
2
, . . . ,
x
n
such that
b
=
n
X
j
=1
a
j
x
j
i.e.,
b
can be written as a
linear combination of the columns of A
.
A compatible system may be regarded as defining an
expansion
or
representation
of
b
in terms of the columns of
A
.
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 staff
 Math, Linear Algebra, Linear Equations, Equations, Systems Of Linear Equations, Vector Space, UCSD Center for Computational Mathematics, UCSD Center, Computational Mathematics

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