---
title: "Bilinear map z = 0,1,∞ to -3,-1,1 with fixed points"
subject: "Applied Mathematics"
chapter: 1
key: "semester_4/applied_mathematics:bilinear-0-1-inf-fixed-points"
source: https://www.bctengineer.com/q/semester_4/applied_mathematics/bilinear-0-1-inf-fixed-points
site: BCT Engineering
---

# Bilinear map z = 0,1,∞ to -3,-1,1 with fixed points

Applied Mathematics · Chapter 1: Complex Analysis

- Asked in: 2071 Magh, 5 marks
- Video: [Bilinear Mapping or Transformation II Mobuis Transformation II Fixed point Problems - Numerical](https://www.youtube.com/watch?v=wfutAHZ-XtI) — Study Buddy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/applied_mathematics/bilinear-0-1-inf-fixed-points
- Whole chapter: https://www.bctengineer.com/md/semester_4/applied_mathematics/chapter-1

## Question

Find the linear transformation which maps the points z = 0, 1, ∞ into the points w = -3, -1, 1 respectively. Find also the fixed points of the transformation.

## Short answer

Use the three point conditions in w=(az+b)/(cz+d). The transformation is w=(z-3)/(z+1). Fixed points satisfy z=(z-3)/(z+1), giving z^2=-3, so fixed points are z=±i√3.

## Answer

Let the transformation be w = (az + b)/(cz + d), with ad − bc ≠ 0.

#### Using the three conditions
- z = ∞ maps to w = 1: a/c = 1, so a = c.
- z = 0 maps to w = −3: b/d = −3, so b = −3d.
- z = 1 maps to w = −1: (a + b)/(c + d) = −1, so a + b = −(c + d).

Substitute a = c and b = −3d:  c − 3d = −c − d, so 2c = 2d and c = d.

So a = c = d. Take a = c = d = 1 and b = −3.

**Answer: w = (z − 3)/(z + 1)**

Check: z = 0 gives −3 ✓. z = 1 gives −2/2 = −1 ✓. z → ∞ gives 1 ✓. Also ad − bc = 1 + 3 = 4 ≠ 0 ✓.

#### Fixed points
Fixed points satisfy w = z:

z = (z − 3)/(z + 1)  →  z² + z = z − 3  →  z² = −3

**Fixed points: z = ± i√3.**

## Check yourself

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

1. z = ∞ mapping to w = 1 gives:
   A. a = −c
   B. b = d
   C. a = d
   D. a = c
   Answer: D. w → a/c as z → ∞. (Key point: ∞ to 1)

2. z = 0 → w = −3 gives:
   A. a = −3c
   B. b = −3d
   C. b = 3d
   D. d = −3b
   Answer: B. b/d = −3. (Key point: 0 to −3)

3. The transformation is:
   A. w = (z + 3)/(z + 1)
   B. w = (z − 3)/(z − 1)
   C. w = (z − 3)/(z + 1)
   D. w = (3z − 1)/(z + 1)
   Answer: C. Check z = 1: −2/2 = −1. (Key point: Map)

4. Fixed points are:
   A. ±3i
   B. ±√3
   C. ±i√3
   D. 1, −3
   Answer: C. z² = −3. (Key point: Fixed points)

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