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

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