Describe what is linear about a linear transformation

Often it is necessary to transform data from one
measurement scale to another. For example, you might want to convert
height measured in feet to height measured in inches. Table 1
shows the heights of four people measured in both feet and inches.
To transform feet to inches, you simply multiply by 12. Similarly,
to transform inches to feet, you divide by 12.

Table 1. Converting between feet and inches.

Feet

Inches

5.00
6.25
5.50
5.75

60
75
66
69

Some conversions require that you multiply by a
number and then add a second number. A good example of this is
the transformation between degrees Centigrade and degrees Fahrenheit.
Table 2 shows the temperatures of 5 US cities in the early afternoon
of November 16, 2002.

Table 2. Temperatures in 5 cities on 11/16/2002.

City

Degrees Fahrenheit

Degrees Centigrade

Houston
Chicago
Minneapolis
Miami
Phoenix

54
37
31
78
70

12.22
2.78
-0.56
25.56
21.11

The formula to transform Centigrade to Fahrenheit
is:

F = 1.8C + 32

The formula for converting from Fahrenheit to
Centigrade is

C = 0.5556F - 17.778

The transformation consists of multiplying by
a constant and then adding a second constant. For the conversion
from Centigrade to Fahrenheit, the first constant is 1.8 and
the second is 32.

Figure 1 shows a plot of degrees Centigrade as a
function of degrees Fahrenheit. Notice that the points form a
straight line. This will always be the case if the transformation
from one scale to another consists of multiplying by one constant
and then adding a second constant. Such transformations are therefore
called linear
transformations.

Figure 1. Degrees Centigrade as a function of degrees Fahrenheit.

Many transformations are not linear. With nonlinear transformations, the points in a plot of the transformed variable against the original variable would not fall on a straight line. Examples of nonlinear transformations are: square root, raising to a power, logarithm, and any of the trigonometric functions.