CO 350 Linear Optimization
TA’s comments for Assignment 2
General:
It seems many students are confused about basic operations in linear algebra. Below we list
some examples of mistakes of these types. Watch out for them.
(i) “If
A
is a matrix and
Ax
= 0 for some vector
x
6
= 0, then
A
= 0.” This is false: take
A
=
1
0
and
x
=
0
1
.
(ii) “If
x
≥
0 is not zero, then
x
is positive.” First of all, you would need to define what it means for
a vector to be positive. If you consider that “
x
is positive” is the same as “
x
i
>
0 for every
i
,”
then this is false: take
x
= (0
,
1)
t
.
(iii) “If
x, y
are vectors with
x
t
y >
0, then
x
and
y
are positive.” This has the same problem of defining
“positive” as before, but this time the statement fails already for real numbers: (

1)
·
(

1) = 1
>
0.
There is also a lot of confusion about the term “unbounded LP.” Consider the following example
(an LP with two variables):
minimize
x
1
subject to
x
1
=
0
0
x
2
=
0
Note that the feasible solutions for this LP are all points in
R
2
of the form (0
, x
2
), where
x
2
is any real
number. Thus, the objective value of each feasible point is
x
1
= 0. So
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 Winter '07
 S.Furino,B.Guenin
 Linear Algebra, Vector Space, basis, TA, linearly independent vectors

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