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1
CHAPTER 7
VECTOR SPACES
There are many mathematical systems in which the notions of
addition and multiplication by scalars are defined.
Example:
R
n
= {all
n
vectors with real entries}.
Example:
R
m
×
n
= {all
m
×
n
matrices with real entries}.
Example:
P
n
= {all polynomials in the variable
x
with degree
≤
n
}.
Example:
P
= {all polynomials in the variable
x
}.
Example:
F
(
R
) = {all functions mapping from
R
to
R
}.
All these mathematical systems share some common fundamental
properties (
axioms
), which may be used to derive a framework
(
general theory
) of new concepts (
definitions
) and useful conclusions
(
theorems
) that apply to each of these mathematical systems.
Definition.
A field
F
is a set in which two operations
“+” (addition)
and “·” (multiplication)
are defined
so that for any
a
,
b
∈
F
, there are
unique elements
a
+
b
(sum of
a
and
b
) and
a
·
b
(product of
a
and
b
)
in
F
, and for all
a
,
b
,
c
∈
F
, the following conditions hold:
1.
a
+
b
=
b
+
a
and
a
·
b
=
b
·
a
(commutativity).
2. (
a
+
b
) +
c
=
a
+ (
b
+
c
) and (
a
·
b
)·
c
=
a
·(
b
·
c
) (associativity).
3.
∃
distinct elements 0 and 1 in
F
such that 0 +
a
=
a
and 1·
b
=
b
(existence of identity elements).
4.
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 Spring '09
 Fong

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