MATH1850U/2050U:
Chapter 4
1
GENERAL VECTOR SPACES
Recall:
In Chapter 3, we saw
n
space or
R
n
.
All together, the following 3 things make
up
n
space:
1.
The objects
2.
Rule for addition
:
a rule for associating with each pair of objects
u
and
v
an
object
v
u
, called the
sum
of
u
and
v
3.
Rule for scalar multiplication:
a rule for associating with each scalar
k
and each
object
u
in
V
an object
u
k
, called the
scalar multiple
of
u
by
k
Real Vector Spaces (4.1; pg. 171)
Definition:
Let
V
be an arbitrary nonempty set of objects on which two operations are
defined, addition and multiplication by scalars (numbers).
If the following axioms are
satisfied by all objects
u
,
v
,
w
in
V
and all scalars
k
and
l
, then we call
V
a
vector space
and we call the objects in
V
vectors
.
1.
If
u
and
v
are objects in
V
, then
v
u
is in
V
.
2.
u
v
v
u
3.
w
v
u
w
v
u
)
(
)
(
4.
There is an object
0
in
V
called the
zero vector
for
V
, such that
u
u
0
0
u
5.
For each
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 Spring '11
 PaulaTu
 Addition, Vector Space, scalar multiplication, KU

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