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1. Range and existence

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: X = country → Y = city, we ask:

  1. What is the capital city of China?
  2. What countries have Beijing as the capital city ?
  3. 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:

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

Motivated by the last question in the example, we define the range of a transformation T: XY to be all the images under the transformation.

rangeT = {T(x): any xX}
= {bY: There is xX, such that T(x) = b}
= {bY: T(x) = b has solutions}

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: RR is all the non-negative numbers. The range of Reflection in x-axis: R2R2 is R2.

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[ x1 ] = [ 3x1 + x2 - x3 ] : R3R3
x2 x1 - x2 + x3
x3 2x1 + 2x2 + x3

In fact, further discussion in this example shows that the range is the whole space R3.

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

T[ x1 ] = [ 3x1 + x2 - x3 ] : R3R3
x2 x1 - x2 + x3
x3 2x1 - x2 + x3

In particular, the range of the transformation is not R3 (strictly smaller than R3).


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