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

- applicability
- The rule applies to any input
`x`∈and produces some output**X**(**T**`x`). - unambiguity
- For any input
`x`∈, the output**X**(**T**`x`) is unique.

Example The transformation `Square`: **R** → **R**, `x` → `x`^{2} 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 `y`^{2} = `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 2^{2} = (-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 `y`^{2} = `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 `S`^{1} in **R**^{2}. The transformation `Sin`: **R** → **R** is given by the following rule:

For any real number `x` ∈ **R**, start from the `b` and move along `S`^{1} in counterclowise direction by distance `x`. Then take the `y`-coordinate of the point we arrive at.