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___________________________________________________________
Linear Algebra: Spring 2007
__________________________________________________________________________________
Proprietary of SAS Lab., Hanyang University  Ansan
61
Chapter 6. Vector spaces
♣
Generalization of Euclidean vectors; we want to construct an arbitrary vector
space, by defining some basic properties to be satisfied.
1. Vector spaces
Definition
Let
V
be a set on which two operations, called
addition
and
scalar multiplication
,
have been defined. If two vectors
u
and
v
are in
V
, the sum of
u
and
v
is denoted by
u
+
v
, and
if
c
is a scalar, the scalar multiple of
u
by
c
is denoted by
c
u
. If the following axioms hold for
all
u
,
v
, and
w
in
V
and for all scalars
c
and
d
, then
V
is called a vector space and its
elements are called vectors;
1.
u
+
v
in
V
(
closed under addition
)
2.
u
+
v
=
v
+
u
(
commutative rule
)
3. (
u
+
v
)+
w
=
u
+(
v
+
w
) (
associative rule
)
4. There exists an element
0
in
V
, called a
zero vector
,
s.t.
u
+
0
=
u
. (
additive identity
)
5. For each
u
in
V
, there is an element –
u
in
V s.t.
u
+(
u
)=
0
. (
additive inverse
)
6.
c
u
in
V
(
closed under scalar multiplication
)
7.
c
(
u
+
v
)=
c
u
+
c
v
(
distributive rule
)
8. (
c+d)
u
=
c
u
+
d
u
(
distributive rule
)
9.
c
(
d
u
)=(
cd)
u
10. 1
u
=
u
╬
By scalars, we mean the real numbers. Accordingly, we should refer
V
as a
vector space over the real numbers.
╬
In fact, the scalars can be any number system (known as a
field
), in which we
can add, subtract, multiply, and divide according to the usual laws of arithmetic.
╬
Roughly, a vector space consists of a set of elements, a field, and vector
operations.
╬
Note the vector operations may be different from our usual arithmetic.
╬
Two most important properties of a vector space is 1 and 6 (
closure
).
For example,
(1) For any
1
≥
n
,
n
ℜ
is a vector space (so called Euclidean vector space) with usual
operations of addition and scalar multiplication.
(2) The set of all
n
m
×
matrices is a vector space with the usual operations of matrix
addition and scalar multiplication. This vector space is denoted
M
mn
.
(3) Let
P
2
denote the set of all polynomials of degree 2 or less with real coefficients; i.e.
{ }
ℜ
∈
+
+
=
2
1
0
2
2
1
0
2
,
,
;
a
a
a
x
a
x
a
a
P
. We define addition and scalar multiplication in the
usual way. If
2
2
1
0
)
(
x
a
x
a
a
x
p
+
+
=
and
2
2
1
0
)
(
x
b
x
b
b
x
q
+
+
=
, then
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View Full Document ___________________________________________________________
Linear Algebra: Spring 2007
__________________________________________________________________________________
Proprietary of SAS Lab., Hanyang University  Ansan
62
2
2
2
1
1
0
0
)
(
)
(
)
(
)
(
)
(
x
b
a
x
b
a
b
a
x
q
x
p
+
+
+
+
+
=
+
2
2
1
0
)
(
x
ca
x
ca
ca
x
cp
+
+
=
(4) Let
F
[
a,b
] denote the set of all real valued functions defined on the closed interval [
a,b
]. If
f
and
g
are two such functions and
c
is a scalar, then
f+g
and
cf
are defined by
)
(
)
(
)
)(
(
x
g
x
f
x
g
f
+
=
+
and
)
(
)
)(
(
x
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This note was uploaded on 06/06/2010 for the course MATH 54 taught by Professor Chung during the Spring '07 term at 카이스트, 한국과학기술원.
 Spring '07
 Chung
 Linear Algebra, Algebra, Vectors, Vector Space

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