3
Linear algebra and basic solutions
To study linear programs, we make crucial use of some basic ideas from linear
algebra. Before proceeding further, we quickly review some of these ideas.
As usual, we use
R
n
to denote the vector space of column vectors with
n
components.
Consider an
m
by
n
matrix
A
.
We say
the columns of
A
span
if the
equation
Ax
=
b
has a solution for every vector
b
in
R
m
. In the language of
linear algebra, we would say, equivalently, that the columns of
A
, which we
denote
A
1
, A
2
, . . . , A
n
, “span” the vector space
R
m
. Introducing additional
columns to a matrix whose columns span does not alter the property.
We say
the columns of
A
are linearly independent
if the only solution
of the equation
Ax
= 0 is
x
= 0.
Again using linear algebra language,
this property is equivalent to the vectors
A
1
, A
2
, . . . , A
n
being distinct and
constituting a “linearly independent set” in
R
m
. Deleting columns from a
matrix whose columns are linearly independent does not alter the property.
Finally, we call the matrix
A
invertible
if its columns both span and are
linearly independent. It’s easy to see that reordering the columns of
A
affects
none of these properties.
Linear algebra teaches us the following important facts. If the columns
of
A
span, then
m
≤
n
. On the other hand, if the columns of
A
are linearly
independent, then
m
≥
n
.
Hence if
A
is invertible, it must be square.
Conversely, if
A
is square, the following properties are equivalent:
•
A
is invertible;
•
the columns of
A
span;
•
the columns of
A
are linearly independent;
•
some matrix
F
satisfies
FA
=
I
(the identity matrix);
•
a sequence of row operations reduces
A
to
I
.
(
Row operations
consist of multiplying a row by a nonzero constant, adding
multiples of rows to other rows, and interchanging rows.) In that case, the
matrix
F
is unique, and is called the
inverse
of
A
, written
A

1
.
It also
satisfies
AA

1
=
I
.
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 Fall '08
 TODD
 Linear Algebra, Vector Space

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