---
title: "Bilinear map of -i, 0, i to -1, i, 1"
subject: "Applied Mathematics"
chapter: 1
key: "semester_4/applied_mathematics:bilinear-minus-i-0-i"
source: https://www.bctengineer.com/q/semester_4/applied_mathematics/bilinear-minus-i-0-i
site: BCT Engineering
---

# Bilinear map of -i, 0, i to -1, i, 1

Applied Mathematics · Chapter 1: Complex Analysis

- Asked in: 2073 Magh, 1+4 marks; 2070 Bhadra, 5 marks; 2070 Magh, 5 marks; 2068 Bhadra (asked 4 times)
- Video: [Bilinear Transformation Cross Ratio Property - Problem 1 - Engineering Mathematics 3](https://www.youtube.com/watch?v=7eOd2KpEkRM) — Ekeeda
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/applied_mathematics/bilinear-minus-i-0-i
- Whole chapter: https://www.bctengineer.com/md/semester_4/applied_mathematics/chapter-1

## Question

Define bilinear transformation. Obtain the bilinear (linear fractional) transformation which maps the points z₁ = -i, z₂ = 0, z₃ = i into w₁ = -1, w₂ = i, w₃ = 1.

## Short answer

A bilinear transformation is w=(az+b)/(cz+d), ad−bc≠0, mapping the extended plane one-to-one. Using the cross-ratio gives w=−i(z−1)/(z+1), equivalently (i−iz)/(z+1). It maps −i,0,i to −1,i,1.

## Answer

#### Definition
A **bilinear (linear fractional, Möbius) transformation** is a mapping

w = (az + b)/(cz + d),  where a, b, c, d are complex constants and ad − bc ≠ 0.

It maps the extended z-plane one-to-one onto the extended w-plane, and it is determined uniquely by three points and their images.

#### Finding the transformation
Use the cross-ratio form with (z₁, z₂, z₃) = (−i, 0, i) and (w₁, w₂, w₃) = (−1, i, 1):

(w − w₁)(w₂ − w₃) / [(w − w₃)(w₂ − w₁)] = (z − z₁)(z₂ − z₃) / [(z − z₃)(z₂ − z₁)]

Right side: (z + i)(0 − i) / [(z − i)(0 + i)] = −(z + i)/(z − i).

Left side: (w + 1)(i − 1) / [(w − 1)(i + 1)]. Since (i − 1)/(i + 1) = i, this equals i(w + 1)/(w − 1).

```text
i (w + 1)/(w − 1) = −(z + i)/(z − i)
(w + 1)/(w − 1)   = i (z + i)/(z − i) = (iz − 1)/(z − i)
```

Write T = (iz − 1)/(z − i). Then w + 1 = T(w − 1), so w = (T + 1)/(T − 1):

```text
w = [(iz − 1) + (z − i)] / [(iz − 1) − (z − i)]
  = (1 + i)(z − 1) / [(i − 1)(z + 1)]
```

Because (1 + i)/(i − 1) = −i,

**Answer: w = −i (z − 1)/(z + 1) = (i − iz)/(z + 1).**

#### Check
- z = −i: w = −i(−i − 1)/(1 − i) = i(1 + i)/(1 − i) = i·i = −1
- z = 0: w = −i(−1)/1 = i
- z = i: w = −i(i − 1)/(i + 1) = −i·i = 1

All three images agree with the given points.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. A bilinear transformation w = (az+b)/(cz+d) requires which condition?
   A. ad + bc = 0
   B. ad − bc ≠ 0
   C. a = d
   D. c = 0
   Answer: B. If ad − bc = 0 the map degenerates to a constant. (Key point: Definition)

2. How many point pairs determine a bilinear transformation uniquely?
   A. One
   B. Four
   C. Three
   D. Two
   Answer: C. Three points and their images fix a, b, c, d up to a common factor. (Key point: Uniqueness)

3. Which relation is used to find the transformation from three point pairs?
   A. Differentiating w with respect to z
   B. Equality of cross-ratios in w and z
   C. Setting w = z
   D. Equating moduli |w| = |z|
   Answer: B. The cross-ratio is invariant, so equating it in both planes gives the map. (Key point: Method)

4. For −i, 0, i → −1, i, 1, the final transformation is:
   A. w = i(z − 1)/(z + 1)
   B. w = −i(z + 1)/(z − 1)
   C. w = −i(z − 1)/(z + 1)
   D. w = (z − 1)/(z + 1)
   Answer: C. Check z=0: −i(−1)/1 = i, and z = i gives −i·(i−1)/(i+1) = 1. (Key point: Result)

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