Game Development Reference
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denote that the mapping F maps a to b by writing F(a) = b (read “F of
a equals b”). Of course, we are primarily interested in the transformations
that can be expressed by using matrix multiplication, but it is important
to note that other mappings are possible.
This section also mentions the determinant of a matrix. We're getting
a bit ahead of ourselves here, since a full explanation of determinants isn't
given until Section 6.1. For now, just know that the determinant of a matrix
is a scalar quantity that is very useful for making certain high-level, shall
we say, determinations about the matrix.
5.7.1
Linear Transformations
We met linear functions informally in Section 4.2. Mathematically, a map-
ping F(
a
) is linear if
F(
a
+
b
) = F(
a
) + F(
b
)
(5.2)
Conditions satisfied if F
is a linear mapping
and
F(k
a
) = kF(
a
).
(5.3)
This is a fancy way of stating that the mapping F is linear if it preserves
the basic operations of addition and multiplication by a scalar. If we add
two vectors, and then perform the transformation, we get the same result
as if we perform the transformation on the two vectors individually and
then add the transformed vectors. Likewise, if we scale a vector and then
transform it, we should get the same resulting vector as when we transform
the vector and then scale it.
There are two important implications of this definition of linear trans-
formation. First, the mapping F(
a
) =
aM
, where
M
is any square matrix,
is a linear transformation, because
F(
a
+
b
) = (
a
+
b
)
M
=
aM
+
bM
= F(
a
) + F(
b
)
Matrix multiplication
satisfies Equation (5.2)
and
F(k
a
) = (k
a
)
M
= k(
aM
) = kF(
a
).
Matrix multiplication
satisfies Equation (5.3)
In other words:
Any transformation that can be accomplished with matrix multiplication
is a linear transformation.
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