You can see why it is called a linear transformation by looking at the plot on the below.

Let b stand for the coefficient each number is multiplied by and A stand for the constant that is added. Y is a linear transformation of X if Y = bX + A Linear transformations are very common.

For example, the Fahrenheit scale of temperature is a linear transformation of the Centigrade scale: F = 1.8 C + 32

A linear transformation is used to convert from tons to pounds:

b = 2000 and A = 0

Pounds = 2000 x Tons + 0

Effects on mean and standard deviation