CSC 240 Homework 3: Transformations I

Introduction

The goal of this homework is to practice working with transformation matrices. Credit: Joe O'Rourke

xkcd
A matrix math joke

Matrix Transform, xkcd

Questions

Please answer the following Yes/No questions.

If Yes, provide a mathematical proof with variables that covers all cases. (Note that providing a numeric example(s) does not suffice to prove that something works in all cases.)
If No, provide a concrete numerical counterexample and a corresponding picture, showing how two particular matrices give different results when multiplied in the opposite order.

Make sure you can multiply matrices by hand, even if you check your work with Mathematica, Matlab, or Wolfram Alpha.

Translations

Do 2D translations commute, i.e., is it always true that T₁T₂ = T₂T₁, where T₁ and T₂ are translation transformations?

Rotations

Do 2D rotations commute, i.e., is it always true that R₁R₂ = R₂R₁, where R₁ and R₂ are rotation transformations?

Translations and Rotations

Do 2D translations commute with rotations, i.e., is it always true that TR = RT, where T is a translation transformation and R is a rotation transformation?

Translations and Scalings

Do 2D translations commute with scalings, i.e., is it always true that TS = ST, where T is a translation transformation and S is a scaling transformation?

Rotations and Scalings

Do rotations commute with scalings, i.e., is it always true that RS = SR, where R is a rotation transformation and S is a scaling transformation?

Submit

If you're interested in learning LaTeX, this might be a good opportunity to practice - you can start from the Transformations Source. Otherwise, write and draw your work, then scan it in as a PDF and submit on Moodle. If you absolutely cannot scan it, you can take pictures, but make sure they are submitted as a single document. Everyone must submit on Moodle.

Collaboration

For this assignment, you are welcome to work together, but everyone should turn in their own work that they have written and understood. Please cite anyone you worked with (and any TAs) at the top of your assignment.