Game Development Reference
InDepth Information
Note also that matrix multiplication is noncommutative. That is, we cannot
say in general that
AB
=
BA
.
1.4.3 Vector Representation and Transformation
We can represent a vector as a matrix with one column, e.g.,
⎡
⎣
⎤
⎦
x
1
x
2
.
x
n
x
=
,
or with one row, e.g.,
b
T
=
b
1
b
m
.
b
2
···
In this topic, we will be using column matrices to represent vectors. Should
we want to represent a row matrix, we shall use the transpose, as above. Using
this notation, we can also represent a matrix as its component columns:
A
=
a
1
a
2
a
n
.
···
is a mapping that preserves the linear properties of
scale and addition; that is, for two vectors
x
and
y
,
A
linear transformation
T
aT
(
x
)+
T
(
y
)=
T
(
a
x
+
y
)
.
We can use matrices to represent linear transformations. Multiplying a vector
x
by an appropriately sized matrix
A
, and expanding the terms, we get
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
b
1
b
2
.
b
m
a
11
a
12
···
a
1
n
x
1
x
2
.
x
n
a
21
a
22
···
a
2
n
=
.
.
.
.
.
.
.
a
m
1
a
m
2
···
a
mn
This represents a linear transformation
from an
n
dimensional space to an
m

dimensional space. If we assume that both spaces use the standard Euclidean
bases
e
1
,
e
2
,...,
e
n
and
e
1
,
e
2
,...,
e
m
, respectively, then the column vectors in
matrix
A
are the transformed basis vectors
T
(
e
n
).
Multiplying transformation matrices together creates a single matrix that rep
resents the composition of the matrices' respective transformations. In this way,
we can represent a composition of linear transformations in a single matrix.
T
(
e
1
)
,
T
(
e
2
)
,...,
T