Databases Reference
In-Depth Information
u
y
b
a
u
x
F I GU R E 12 . 2
Example of different vectors.
By
vector addition
of two vectors, we mean the vector obtained by the pointwise addition
of the components of the two vectors. For example, given two vectors
a
and
b
:
⎡
⎣
⎤
⎦
,
⎡
⎣
⎤
⎦
a
1
a
2
a
3
b
1
b
2
b
3
=
=
(1)
a
b
the vector addition of these two vectors is given as
⎡
⎤
a
1
+
b
1
⎣
⎦
a
+
b
=
a
2
+
b
2
(2)
a
3
+
b
3
By
scalar multiplication
, we mean the multiplication of a vector with a real or complex
number. For this set of elements to be a vector space it has to satisfy certain axioms.
Suppose
V
is a vector space;
x
,
y
,
and
z
are vectors; and
α
and
β
are scalars. Then the
following axioms are satisfied:
1.
x
+
y
=
y
+
x
(commutativity).
(
x
+
y
)
+
z
=
x
+
(
y
+
z
)
and
(αβ)
x
=
α(β
x
)
(associativity).
2.
3.
There exists an element
θ
in
V
such that
x
+
θ
=
x
for all
x
in
V
.
θ
is called the additive
identity.
α(
x
+
y
)
=
α
x
+
α
y
, and
(α
+
β)
x
=
α
x
+
β
x
(distributivity).
4.
5.
1
·
x
=
x
, and 0
·
x
=
θ
.
.
A simple example of a vector space is the set of real numbers. In this set zero is the
additive identity. We can easily verify that the set of real numbers with the standard operations
of addition and multiplication obey the axioms stated above. See if you can verify that the set
of real numbers is a vector space. One of the advantages of this exercise is to emphasize the
fact that a vector is more than a line with an arrow at its end.
6.
For every
x
in
V
, there exists a
(
−
x
)
such that
x
+
(
−
x
)
=
θ
