Game Development Reference
In-Depth Information
Definition 2.11. A set of n vectors
ee
,
,
is linearly independent if
,
{
}
12
n
aa
,
,
, where at least one of the a is not
,
there do not exist real numbers 12
n
zero, such that
e
e
e
0
.
(2.40)
a
+
a
+
+
a
=
11
2 2
nn
Otherwise, the set
ee
,
,
is called linearly dependent .
,
{
}
12
n
An n -dimensional vector space is one that can be generated by a set of n line-
arly independent vectors. Such a generating set is called a basis, whose formal
definition follows.
Definition 2.12. A basis for a vector space V is a set of n linearly independ-
ent vectors
=
{
ee
,
,
,
e
}
for which, given any element P in V , there exist
12
n
aa
,
,
such that
,
real numbers 12
n
Pe
=+ ++
a
a
e
a
e
.
(2.41)
11
2 2
nn
Every basis of an n -dimensional vector space has exactly n vectors in it. For in-
stance, it is impossible to find a set of four linearly independent vectors in ,
and a set of two linearly independent vectors is insufficient to generate the entire
vector space.
There are an infinite number of choices for a basis of any of the vector spaces
 . We assign special terms to those that have certain properties.
Definition 2.13. A basis
=
ee
,
,
,
e
for a vector space is called orthog-
{
}
12
n
()
onal if for every pair
, ij with i
j
, we have
e
=
0
.
e
i
j
The fact that the dot product between two vectors is zero actually implies that the
vectors are linearly independent, as the following theorem demonstrates.
Theorem 2.14. Given two nonzero vectors e and e , if 12 0
e
=
, then e and
e
e are linearly independent.
Proof. We suppose that e and e are not linearly independent and arrive at a con-
tradiction. If e and e are linearly dependent, then there exist scalars a and a