CO350 Assignment 1
–
Alex Fok
20300650
,
May 14
2010
:

Instructor
Mathieu Guay Paquet
( )
1 a
Given
=
,
Ax
b has a solution
s
Columns of A are not linearly independent
Proof
=
Let K be the solution set of the system Ax
b
=
Let KH be the solution set of the corresponding homogenous system Ax
0
.
Let LA denote the left multiplication transformation

Let ai denote the i th column of A
,
Since the columns of A are not linearly independent
=
,…,
>
rankA
dimspana1
an n
=
(
).
,
Note that KH
N LA
By dimension theorem
= 
= 
>  =
dimKH
n rankLA
n rankA n n
0
,

∈
.
Therefore
there exists a non trivial solution u
KH
+
∈
,
Consider s tu for t
R
+
=
+
=
=
As tu
As tAu
As
b
, +
∈
.
,
.
Hence
s tu
K
That is
there are infinitely many solutions
( )
1 b
Given
=
,
Ax
b has a solution
s
Columns of A are linearly independent
Proof
( ).
Using the same notation in part a
,
Since the columns of A are linearly independent
=
,…,
=
rankA
dimspana1
an
n
=
(
).
,
Note that KH
N LA
By dimension theorem
= 
= 
=  =
dimKH
n rankLA
n rankA
n n
0
,
.
= +
={ }.
Therefore
KH is trivial
K
s KH
s
,
=
.
Thus
Ax
b has a unique solution
2
(
:
= , ≥ )
,
Suppose max ctx Ax
b x
0
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 Fall '07
 S.Furino,B.Guenin
 Optimization, basis, Existence, infinitely many solutions, Solution Set, Dimension Theorem

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