Linear Transformations

Prerequisites
None

Learning Objectives

1. Give the formula for a linear transformation
2. Determine whether a transformation is linear
3. 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. To transform feet to inches, you simply multiply by 12.

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. The formula to transform Centigrade to Fahrenheit is:

F = 1.8C + 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 