Complex Analysis - Maths KU

Note: The order in which you perform
the steps in a linear mapping
is important!
☞
2.3 Linear Mappings 73
linear functions is again a linear function, it follows that the composition of
finitelymanyrotations, magnifications, and translations is a linear mapping.
We have seen that translations, rotations, and magnifications all preserve
the basic shape of a figure in the complex plane. A linear mapping, therefore,
will also preserve the basic shape of a figure in the complex plane. This
observation is an important propertyof complex linear mappings and is worth
repeating.
A complex linear mapping w = az + b with a �= 0can distort the size of a
figure in the complex plane, but it cannot alter the basic shape of the figure.
When describing a linear function as a composition of a rotation, a magnification,
and a translation, keep in mind that the order of composition is
important. In order to see that this is so, consider the mapping f(z) =2z + i,
which magnifies by2, then translates byi; so, 0 maps onto i under f. If we
reverse the order of composition—that is, if we translate by i, then magnifyby
2—the effect is 0 maps onto 2i. Therefore, reversing the order of composition
can give a different mapping. In some special cases, however, changing the
order of composition does not change the mapping. See Problems 27 and 28
in Exercises 2.3.
A complex linear mapping can always be represented as a composition
in more than one way. The complex mapping f(z) =2z + i, for example,
can also be expressed as f(z) =2(z + i/2). Therefore, a magnification by2
followed bytranslation byi is the same mapping as translation by i/2 followed
bymagnification by2.
EXAMPLE 4 Image of a Rectangle under a Linear Mapping
Find the image of the rectangle with vertices −1+i, 1+i, 1+2i, and −1+2i
under the linear mapping f(z) =4iz +2+3i.
Solution Let S be the rectangle with the given vertices and let S ′ denote the
image of S under f. We will plot S and S ′ in the same copyof the complex
plane. Because f is a linear mapping, our foregoing discussion implies that
S ′ has the same shape as S. That is, S ′ is also a rectangle. Thus, in order
to determine S ′ , we need onlyfind its vertices, which are the images of the
vertices of S under f:
f(−1+i) =−2 − i f(1 + i) =−2+7i
f(1+2i) =−6+7i f(−1+2i) =−6 − i.
Therefore, S ′ is the rectangle with vertices −2−i, −2+7i, −6+7i, and −6−i.
The linear mapping f(z) =4iz +2+3i in Example 4 can also be viewed
as a composition of a rotation, a magnification, and a translation. Because
Arg(4i) =π/2 and |4i| =4,f acts byrotating through an angle of π/2 radians
about the origin, magnifying by 4, then translating by 2 + 3i. This sequence