We have introduced some concepts for transformations in the process of developing lienar algebra. Now we discuss general transformations in more detail.
Transformations have been defined as rules that convert one data into another data. In fact, this is not quite rigorous as a definition. The following is more precise.
A transformation T from a set X (called source or domain) to another set Y (called target or codomain) is a rule that assigns for x ∈ X an element (called image or value) T(x) ∈ Y. The rule should be well-defined in the following sense.
Example The transformation Square: R → R, x → x2 is given by the following rule:
For any real number x ∈ R, multiply x to itself.
Note that multiplying to itself is a process that can be applied to any real number. Moreover, the result is unambiguous, such as the square of -2 is 4, not any other number. The transformation is well-defined.
Example Consider the rule Square Root: R → R given by the following:
For any x ∈ R, find a number y ∈ R, such that y2 = x.
Unfortunately, the rule cannot be applied to the negative numbers. Moreover, even if we can apply the rule to non-negative numbers, such as 4, the result is ambiguous because we have 22 = (-2)2 = 4, for example. Thus both conditions for the transformation fail, and Square Root: R → R is not a transformation.
To make the rule well-defined, we first need to restrict ourself to non-negative numbers and consider √: [0, ∞) → R. Moreover, we make a specific choice among the two positive candidates for y with the following modified rule:
For any x ∈ [0, ∞), find a number y ∈R, such that y2 = x and y ≥ 0.
Then √ is well-defined and is the usual square root function.
Example Let b = (1, 0) be a base point of the unit circle S1 in R2. The transformation Sin: R → R is given by the following rule:
For any real number x ∈ R, start from the b and move along S1 in counterclowise direction by distance x. Then take the y-coordinate of the point we arrive at.