Section 1.2
Solid Mechanics Part III
Kelly
10
1.2 Vector Spaces
The notion of the vector presented in the previous section is here recast in a more formal
and abstract way.
This might seem at first to be unnecessarily complicating matters, but
this approach turns out to be helpful in unifying and bringing clarity to much of the theory
which follows.
Some background theory which complements this material is given in Appendix A to this
Chapter, §1.A.
1.2.1
The Vector Space
The vectors introduced in the previous section obey certain rules, those listed in §1.1.3.
It
turns out that many other mathematical objects obey the same list of rules.
For that
reason, the mathematical structure defined by these rules is given a special name, the
linear space
or
vector space
.
First, a
set
is any welldefined list, collection, or class of objects, which could be finite or
infinite.
An example of a set might be
{ }
3

≤
=
x
x
B
(1.2.1)
which reads “
B
is the set of objects
x
such that
x
satisfies the property
3
≤
x
”.
Members
of a set are referred to as
elements
.
Consider now the
field
1
of real numbers
R
.
The elements of
R
are referred to as
scalars
.
Let
V
be a nonempty set of elements
K
,
,
,
c
b
a
with rules of
addition
and
scalar
multiplication
, that is there is a
sum
V
∈
+
b
a
for any
V
∈
b
a
,
and a
product
V
∈
a
α
for any
V
∈
a
,
R
∈
.
Then
V
is called a
(real
)
2
vector space
over
R
if the following
eight axioms hold:
1.
associative law for addition
: for any
V
∈
c
b
a
,
,
, one has
)
(
)
(
c
b
a
c
b
a
+
+
=
+
+
2.
zero element
: there exists an element
V
∈
o
, called the zero element, such that
a
a
o
o
a
=
+
=
+
for every
V
∈
a
3.
negative
(or
inverse
): for each
V
∈
a
there exists an element
V
∈
−
a
, called the
negative of
a
, such that
0
)
(
)
(
=
+
−
=
−
+
a
a
a
a
4.
commutative law for addition
: for any
V
∈
b
a
,
, one has
a
b
b
a
+
=
+
5.
distributive law, over addition of elements of V
: for any
V
∈
b
a
,
and scalar
R
∈
,
b
a
b
a
+
=
+
)
(
6.
distributive law, over addition of scalars
: for any
V
∈
a
and scalars
R
∈
β
,
,
a
a
a
+
=
+
)
(
1
A
field
is another mathematical structure (see Appendix A to this Chapter, §1.A).
For example, the set of
complex numbers is a field.
In what follows, the only field which will be used is the familiar set of real
numbers with the usual operations of addition and multiplication.