2.1 A Linear Map

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I have written a program which draws some random dots inside the square

fx + iy 2 C : 􀀀1 _ x _ 1;􀀀1 _ y _ 1g

which is shown in _gure 2.1.

The second _gure 2.2 shows what happens when each of the points is multiplied

by the complex number 0:7 + i0:1. The set is clearly stretched by a

number less than 1 and rotated clockwise through a small angle.

This is about as close as we can get to visualising the map

w = (0:7 + i0:1)z

This is analogous to, say, y = 0:7x, which shrinks the line segment [-1,1]

down to [-0.7,0.7] in a similar sort of way. We don't usually think of such a

map as shrinking the real line, we usually think of a graph.

2.1. A LINEAR MAP 35

Figure 2.2: After multiplication

Figure 2.3: After multiplication and shifting

36 CHAPTER 2. EXAMPLES OF COMPLEX FUNCTIONS

And this is about as simple a function as you could ask for.

For a slightly more complicated case, the next _gure 2.3 shows the e_ect of

w = (0:7 + i0:1)z + (0:2􀀀i0:3)

which is rather predictable.

Functions of the form f(z) = wz for some _xed w are the linear maps from C

to C . Functions of the form f(z) = w1z+w2 for _xed w1; w2 are called a_ne

maps. Old fashioned engineers still call the latter 'linear'; they shouldn't.

The distinction is often important in engineering. The adding of some constant

vector to every vector in the plane used to be called a translation. I

prefer the term shift. So an a_ne map is just a linear map with a shift.

The terms 'function', 'transformation', 'map', 'mapping' all mean the same

thing. I recommend map. It is shorter, and all important and much used

terms should be short. I shall defer to tradition and call them complex

functions much of the time. This is shorter than 'map from C to C ', which is

necessary in general because you do need to tell people where you are coming

from and where you are going to.

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