Transformations

Section 4 Transformations

It is quite possible that only one set be involved with a function. The only way that this can happen is that images of all elements be back in the same set. The following definition makes this clear.

[4.1] A transformation on a set is a function whose range is contained in that set.

To the right is a figure showing an intuitive idea of a transformation. T sends elements of a set (domain) onto a subset (range).

Often the capital letters S and T are used as transformations in these discussions. Sometimes R is used when there is no confusion between transformations and the set of real numbers. The capital letter I still denotes the identity transformation.

Click here to see examples of transformations, their products and inverses. Some of those discussed there are not one-to-one correspondences.

Because only one set is involved the range of any transformation is contained in that set which is also the domain. Therefore, it is always possible to form products ST and TS of any two transformations S and T on the same set. There is no problem of not being conformable to each other. However, the products may carry some elements of the set onto different images, and therefore ST and TS are not always equal.

Suppose S and T are transformations on the set N₃ = {1,2,3} and defined by

S(1) = 2,  S(2) = 3,  S(3) = 1     T(1) = 2,  T(2) = 1,  T(3) = 3 Then TS(1) = T(2) = 1    TS(2) = T(3) = 3     TS(3) = T(1) = 2
ST(1) = S(2) = 3     ST(2) = S(1) = 2    ST(3) = S(3) = 1 The arrow notation shows more clearly that TS and ST do different things to the elements 1,2,3:
    S        T              TS        ST
    1->2    1->2          1->1    1->3
    2->3    2->1          2->3    2->2
    3->1    3->3          3->2    3->1 The operation of forming products of transformations is not commutative.

A positive real number x has powers: x² = xx, x³ = xxx. ... . Similarly, a transformation T has powers: T² = TT, T³ = TTT,... . T¹ = T. These powers of T do commute: T²T³ = T⁵ = T³T².

Another situation where transformations do commute is a product of two transformations in which one (or both) is the identity transformation. TI = IT because both sides of this equation are equal to T itself.

Still another situation of commutivity is the following:

[4.2] If T is a transformation on a set, and T is a one-to-one correspondence then both the products

T*T^-1 and T^-1*T are equal to the identity transformation.

If T carries an element x onto an element y then by definition the inverse T^-1 carrys y back onto x. Therefore,

T^-1*T(x) = T^-1(y) = x = I(x) Since this is true for all elements x in the set, T^-1*T = I.
Now let y be any element of the set. Then let x = T^-1(y). Then T*T^-1(y) = T(x) = y = I(y). Since this is true for all elements y in the set, T*T^-1 = I.

This motivates the definition T° = I.

Looking at the arrow definition of T in the example before [4.2] above, element 3 is not carried onto another element. Intuitively speaking, 3 is its own image and is not "moved" by T.

[4.3] Under a transformation if the image of an element is that same element, then that element is a fixed point of the transformation. The transformation is said to leave that element fixed.

If T(a) = a the a is a fixed point, and T leaves a fixed. Another way of saying it, a is its own image T(a).

[4.4] On any set the identity transformation I and only the identity transformation I leaves fixed every element in the set.

In the relation-set of the identity transformation on any set, the first coordinate of each ordered pair is the same as the second coordinate. Also all the elements of the set appear as first coordinates once and only once. Therefore, all the elements appear as second coordinates once and only once. This means that the identity transformation is a one-to-one correspondence between the set and itself. In the above example each of the four transformations S, T, ST, TS is a one-to-one correspondence between the set N₃ and itself.

The transformation on the set of integers J that carries each integer onto its negative is a one-to-one correspondence between J and itself.

[4.5a] A finite permutation is a transformation on a finite set, and that transformation is also a one-to-one correspondence between the set and itself.
[4.5b] An infinite permutation is a transformation on an infinite set, and that transformation is also a one-to-one correspondence between the set and itself.
[4.5c] A general permutation is a transformation on any set, and that transformation is also a one-to-one correspondence between the set and itself.

The transformations S, T, ST, TS in the above example are finite permutations. The transformation on J that carries integers onto their negatives is an infinite permutation. The reason for the terms "finite" and "infinite" and "general" is that in most places the single term "permutation" means in these discussions "finite permutation". A one-to-one correspondence between a finite set and itself is called a finite permutation

In a room the furniture is rearranged. But the new arrangement is not liked, so all the furniture is moved back to original positions. Two actions have occurred. The second action undoes the first action. The second action is the opposite or, more technically, the inverse of the first action.

On the set J of integers, a transformation T adds 5 to each integer: T(x) = x + 5. Another transformation S does just the opposite, it subtracts 5 from each integer: S(x) = x - 5. The product ST leaves all integers as they were: ST(x) = S(x + 5) = (x + 5) - 5 = x.

On the set R⁺ of positive real numbers multiplication and division are possible. If T multiplies every positive real number by 5 (T(x)=x*5) then the transformation S(x)=x/5, which divides each positive real number by 5, is the inverse of T.

[4.6] A transformation on a set is an inverse of a general permutation on the same set if and only if their product is the identity transformation.

In symbols, if ST = I then S is the inverse of T.

Recall that the product of two transformations may or may not be commutative. The definition [4.6] does not indicate the order of the product. If S is the inverse of T then is TS = I also true? The answer is yes.

A very important category of transformations are geometric transformations on (of) the plane. These are discussed in more detail in another volume of "Intuitive Mathematics." But among them are the congruence transformations that move geometric figures around without distortion that changes size or shape. (They are also called distance preserving transformations or isometries.)

Consider an equilateral triangle. It is completely determined by its vertices, labeled here a,b,c. These are points (elements) of the set of points that make up the triangle. There is a center o which is not on the triangle, but is located by the intersections of medians, altitudes and angle-bisectors.

Let R be a transformation on the set {a,b,c} defined by:

R
a ---> b
b ---> c
c ---> a Now translate R into movement of the triangle. Move vertex a onto b. At the same time move vertex b onto c and c onto a. This can be done by picking up the triangle and rotating it 120° in a counter-clockwise rotation.

Consider R². It is the product of R with itself. Then

RR(a) = R(b) = c
RR(b) = R(c) = a
RR(c) = R(a) = b This means
R²
a ---> c
b ---> a
c ---> b Therefore, R² rotates the triangle 240° around the center in a counter-clockwise direction.
R³ = product RRR. It is easily shown that RRR
a ---> a
b ---> b
c ---> c Since R³ leaves every vertex fixed, it is equal to the identity transformation I. In general, Rⁿ is a rotation through 120n degrees, for any positive integer n.

This can be extended to all integers n. Notice that RR² = RRR = I. Therefore by [ If n < 0 then rotate the triangle in the opposite direction -120n degrees, that is clockwise. To complete the expansion to integers, it is natural to define R⁰ = I, and R¹ = R.

By means of the exponents on R, there is a natural one-to-one correspondence between the set J = {0,1,2} of integers mod 3 and the set of the three rotations {R⁰,R¹,R²}

0 <---> I⁰
1 <---> R
2 <---> R²