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Second, any linear transformation will transform the zero vector into
the zero vector. If F(
0
) =
a
,
a
=
0
, then F cannot be a linear mapping,
since F(k
0
) =
a
and therefore F(k
0
) = kF(
0
). Because of this,
Linear transformations do not contain translation.
Since all of the transformations we discussed in Sections 5.1 through 5.5
can be expressed using matrix multiplication, they are all linear transfor-
mations.
In some literature, a linear transformation is defined as one in which par-
allel lines remain parallel after transformation. This is almost completely
accurate, with two exceptions. First, parallel lines remain parallel after
translation, but translation is not a linear transformation. Second, what
about projection? When a line is projected and becomes a single point, can
we consider that point “parallel” to anything? Excluding these technicali-
ties, the intuition is correct: a linear transformation may “stretch” things,
but straight lines are not “warped” and parallel lines remain parallel.
5.7.2 Affine Transformations
An a
ne transformation is a linear transformation followed by translation.
Thus, the set of a
ne transformations is a superset of the set of linear
transformations: any linear transformation is an a
ne translation, but not
all a
ne transformations are linear transformations.
Since all of the transformations discussed in this chapter are linear trans-
formations, they are all also a
ne transformations (though none of them
have a translation portion). Any transformation of the form
v
′
=
vM
+
b
is an a
ne transformation.
5.7.3 Invertible Transformations
A transformation is invertible if there exists an opposite transformation,
known as the inverse of F, that “undoes” the original transformation. In
other words, a mapping F(
a
) is invertible if there exists an inverse mapping
F
−1
such that
−1
(F(
a
)) = F(F
−1
(
a
)) =
a
F
−1
is also invertible.
There are nona
ne invertible transformations, but we will not consider
them for the moment. For now, let's concentrate on determining if an a
ne
transformation is invertible. As already stated, an a
ne transformation
for all
a
. Notice that this implies that F
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