The left side of a system of linear equations has been considered as a transformation. Now we raise some natural questions about transformations.

Example For the transformation
` T = Capital City`:

- What is the capital city of China?
- What countries have Beijing as the capital city ?
- Is there a country with Beijing as the capital? Is any city the capital of some country?

In mathematical terminology, the questions can be rephrased as:

- Find the value of
= China ∈**x**under**X**.**T** - What
∈**x**has**X**= Beijing ∈**b**as the value under**Y**? Equivalently, find**T**satisfying**x**(**T**) =**x**.**b** - Is
= Beijing ∈**b**the value of some**Y**∈**x**under**X**? Equivalently, do the equation**T**(**T**) =**x**have solutions for**b**= Beijing? What about any city**b**?**b**

Motivated by the last question in the example, we define the range of a transformation
` T`:

`range T`
= {

= {

= {

Example The range of the transformation

`Capital City`: country → city

is all the capital cities in the world. For example, Beijing is an element of `range`(`Capital City`),
while Hong Kong is not an element of `range`(`Capital City`).

The range of the transformation

`Instructor`: course → professor

is all the professors who teach at least one course.
For example, I belong to the set `range`(`Instructor`),
while the president of the university (who is so busy) is unlikely an element of `range`(`Instructor`).

As for the transformation

`Age`: people → number

we know 20 is in `range`(`Age`) because there are millions of people at age 20.
We are also pretty sure 100 is in `range`(`Age`), although we are less sure about 110.
On the other hand, we are absolutely certain that 200 and -10 are not in `range`(`Age`).

Example
The range of `Square`: **R** → **R** is all the non-negative numbers.
The range of `Reflection in x-axis`: **R**^{2} → **R**^{2} is **R**^{2}.

Since the system in this example has solutions, the vector (2, 2, 6) (which is the right side of the system) is in the range of

[T |
x_{1} |
] = [ | 3x_{1} + x_{2} - x_{3} |
] : R^{3} → R^{3} |

x_{2} |
x_{1} - x_{2} + x_{3} |
|||

x_{3} |
2x_{1} + 2x_{2} + x_{3} |

In fact, further discussion in this example
shows that the range is the whole space **R**^{3}.

In contrast, this example shows that (2, 2, 6) is not in the range of

[T |
x_{1} |
] = [ | 3x_{1} + x_{2} - x_{3} |
] : R^{3} → R^{3} |

x_{2} |
x_{1} - x_{2} + x_{3} |
|||

x_{3} |
2x_{1} - x_{2} + x_{3} |

In particular, the range of the transformation is not **R**^{3}
(strictly smaller than **R**^{3}).