# Linear Fractional Transformation - find the formula

• anniecvc
In summary: Therefore, in summary, the linear fractional formula for finding the "red point" f(x) is f(x)=\frac{a(x-1)}{c(x+d)}, where a and c are chosen arbitrarily and b=-a.
anniecvc
http://math.sfsu.edu/federico/Clase/Math350.S15/linea.JPG 1. Homework Statement
The picture below represents the map from a "green projective line" to a "red projective line." It takes the "green points" 1,3,7,-11 to the "red points" 0,6,10,20, respectively as shown by the ruler. Let f be the corresponding linear fractional, so f(1) = 0, f(3) = 6, f(7) = 10, f(-11) = 20. Find a formula for the "red point" f(x) on the ruler where the "green point" x lands.

## Homework Equations

a linear fractional is where f(x) = (ax+b)/(cx+d) where ad-bc is not equal to 0.

## The Attempt at a Solution

First a did a system of equations:
(a(1) + b) / ( c(1) + d ) = 0
(a(3) +b) / ( c(3) + d )= 6
(a(7) +b) / ( c(7) + d ) = 10
(a(-11) +b) / (c(-11) + d )= 20

Thinking I could solve for a,b,c,d. However if I get all the letters in terms of let's say d and solve, this becomes nonsensical, since something d over something d is a number (the d's reduce) , and I'm left with a number = another number.

Last edited by a moderator:
anniecvc said:
a linear fractional is where f(x) = (ax+b)/(cx+d) where ad-bc is not equal to 0.
Since you have a fraction, you can choose either a or c to be 1. So, choose c = 1. The first equation says a⋅1 + b = 0, so b = -a. That's two variables accounted for. Now the rest is easy...

Choosing a or c to be 1 (or any number) seems to be an invalid assumption since even this is a ratio of a sum not just of a number over a number (choosing c=1 affects d, and choosing a=1 affects b), but please explain if I am incorrect. I chose c to be 1, then since I had everything else worked out in terms of d I plugged it all in. When I did, c=1 did not give f(3) = 6, so I solved for what c would be and got c = 1/6. Then, I tried to test this formula with f(7) = 10. It didn't work.

anniecvc said:
Choosing a or c to be 1 (or any number) seems to be an invalid assumption since even this is a ratio of a sum not just of a number over a number (choosing c=1 affects d, and choosing a=1 affects b), but please explain if I am incorrect. I chose c to be 1, then since I had everything else worked out in terms of d I plugged it all in. When I did, c=1 did not give f(3) = 6, so I solved for what c would be and got c = 1/6. Then, I tried to test this formula with f(7) = 10. It didn't work.
I am sorry, but you have made an error somewhere. One solution is: a = 15, b= -15, c=1, d=2.

About choosing one of the variables: A fundamental fact of fractions is that you can multiply or divide by the same number above or below. Therefore $\frac{a\cdot x + b}{c\cdot x + d}=\frac{a(x+\frac{b}{a})}{a(\frac{c\cdot x}{a}+\frac{d}{a})}=\frac{c(\frac{a}{c}x+\frac{b}{c})}{c(x+\frac{d}{c})}$ (of course, you cannot do that if the one you choose turns out to be 0).

Thanks very much Svein.

Another quick simplification is to notice the case where f(x)=0

$$\frac{a+b}{c+d}=0$$
and for a fraction to equal 0, it means that the numerator = 0, hence

$$a+b=0$$

$$a=-b\text{, or } b=-a$$

Therefore you just have an equation of the form

$$f(x)=\frac{ax-a}{cx+d}$$

## 1. What is a linear fractional transformation?

A linear fractional transformation, also known as a Möbius transformation, is a mathematical function that maps one complex number to another using a linear equation. It is represented by the formula f(z) = (az + b)/(cz + d), where a, b, c, and d are complex numbers and z is the input variable.

## 2. How do I find the formula for a linear fractional transformation?

The formula for a linear fractional transformation can be found by identifying the values of a, b, c, and d in the equation f(z) = (az + b)/(cz + d). These values can be determined by knowing the transformation's fixed points (points that do not change after the transformation) and by solving a system of equations with these points.

## 3. What are the properties of a linear fractional transformation?

A linear fractional transformation has several important properties, including linearity (meaning it preserves lines and straightness), conformality (meaning it preserves angles), and invertibility (meaning it has an inverse function). It also maps circles and lines to circles and lines, and can be composed with other linear fractional transformations.

## 4. Can linear fractional transformations be used for any type of numbers?

Yes, linear fractional transformations can be used for both real and complex numbers. In fact, they are commonly used in complex analysis to study the behavior of functions on the complex plane. They are also used in other areas of mathematics, such as geometry and algebra.

## 5. What are some applications of linear fractional transformations?

Linear fractional transformations have many practical applications, including in computer graphics, image processing, and control theory. They can also be used to solve various mathematical problems, such as finding the inverse of a function or determining the symmetry of a geometric figure.

Replies
8
Views
502
Replies
9
Views
2K
Replies
7
Views
1K
Replies
1
Views
578
Replies
0
Views
576
Replies
3
Views
1K
Replies
1
Views
1K
Replies
10
Views
1K
Replies
11
Views
2K
Replies
3
Views
802