Chapter 5
General Vector Spaces
1
Introduction
Let
A
be an
m
×
n
matrix and let
b
∈
R
m
. Then we have a linear system
Ax
=
b
(1)
with variables
x
= (
x
1
,...,x
n
)
T
∈
R
n
.
So far in this course we have developed methods to
analyze and solve such systems. In the remainder of the course we develop methods to reduce
more general types of problems to linear systems of this form.
This will allow us to use our
knowledge of linear systems to solve a much broader and more interesting class of mathematical
problems.
Let
V
and
W
be sets and let
T
:
V
→
W
be a function (“transformation”). In this setup we
can consider the problem of solving
T
(
f
) =
g
(2)
where
g
is a given element of
W
and
f
is a variable element of
V
which we wish to solve for. We
are going to find general conditions on
V
,
W
and
T
which allow (2) to be reduced to a linear
system of the form (1), which we can then solve using our knowledge of linear systems.
The
specific conditions are that
V
and
W
should be
general vector spaces
(with finite dimensions),
and
T
should be a
linear transformation
. We discuss general vector spaces in this chapter and
linear transformations in the following chapter 6.
2
General Real Vector Spaces: Definition and Examples
2.1
Motivation
The Euclidean
n
space
R
n
is a set on which addition and scalar multiplication are defined:
(
u
1
,...,u
n
) + (
v
1
,...,v
n
)
=
(
u
1
+
v
1
,...,u
n
+
v
n
)
c
(
u
1
,...,u
n
)
=
(
cu
1
,...,cu
n
)
Addition and scalar multiplication on
R
n
satisfy certain properties: associative law for vector
addition, distributive laws, there is a zero vector, multiplying a vector with the scalar 1 has no
effect etc.
Aside from
R
n
, we can also consider
other sets
on which addition and scalar multiplication are
defined and have the same properties. Such sets are call “real vector spaces”.
2.2
Definition
A
real vector space
is an nonempty set
V
on which addition and scalar multiplication are
defined (this means that for any
u,v
∈
V
and
c
∈
R
it must be defined what
u
+
v
and
cu
are)
such that the following conditions are satisfied for all
u,w,v
∈
V
and all
c,d
∈
R
.
1.
u,w
∈
V
⇒
u
+
w
∈
V
(closed under addition)
2.
u
∈
V
,
c
∈
R
⇒
cu
∈
V
(closed under scalar multiplication)
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3. There is an element 0
∈
V
such that
u
+ 0 = 0 +
u
=
u
for all
u
∈
V
(existence of zero
vector)
4. For every
u
∈
V
there is an element
−
u
∈
V
with
u
+ (
−
u
) = 0 (existence of negative
vector)
5. (
u
+
v
) +
w
=
u
+ (
v
+
w
) (associative law for addition)
6.
u
+
v
=
v
+
u
(commutative law for addition)
7.
c
(
u
+
v
) =
cu
+
cv
(distributive law for vector addition)
8. (
c
+
d
)
u
=
cu
+
du
(distributive law for addition of scalars)
9.
c
(
du
) = (
cd
)
u
(associative law for scalar multiplication)
10. 1
u
=
u
(multiplication with scalar 1 has no effect)
In the following, we will often simply refer to real vector spaces as “vector spaces”.
2.3
Examples of Real Vector Spaces
Euclidean
n
spaces and subspaces thereof
•
Let
n
be any positive integer. Then
R
n
is a real vector space. We have verified already
that the conditions 110 hold for
V
=
R
n
.
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 Spring '10
 ProfDavidH.Adams
 Linear Algebra, Vector Space

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