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VECTOR SPACES
Many concepts concerning vectors in
R
n
can be extended to other
mathematical systems.
We can think of a
vector space
in general, as a collection of objects
that behave as vectors do in
R
n
. The objects of such a set are called
vectors
.
1. Vector Space
A
vector space
is a nonempty set
V
of objects, called
vectors
, on
which are deﬁned two operations, called
addition
and
multiplication by
scalars
(real numbers), subject to the ten axioms below. The axioms
must hold for all
u
,
v
and
w
in
V
and for all scalars
k
and
m
.
1.
u
+
v
is in
V
.
2.
u
+
v
=
v
+
u
.
3. (
u
+
v
) +
w
=
u
+ (
v
+
w
)
4. There is a vector (called the zero vector)
0
in
V
such that
u
+
0
=
u
.
5. For each
u
in
V
, there is vector

u
in
V
satisfying
u
+ (

u
) =
0
.
6.
k
u
is in
V
.
7.
k
(
u
+
v
) =
k
u
+
k
v
.
8. (
k
+
m
)
u
=
k
u
+
m
u
.
9. (
km
)
u
=
k
(
m
u
)
.
10. 1
u
=
u
.
1
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The Zero Vector Space
Let
V
consist of a single object, which denote by
0
, and deﬁne:
0+0=0
and
k
0+0=0
for all skalar
k
.
It is easy to check that all vector space axioms are satisﬁed.
This is called the
zero vector space.
Some properties of Vectors
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This note was uploaded on 06/20/2008 for the course AM 025b taught by Professor Kiruscheva during the Winter '07 term at UWO.
 Winter '07
 KIRUSCHEVA

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