Point Mapping Lab

Lesson 1: Matrices as Linear Transformations

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The Transformation Function

In linear algebra, a matrix acts as a function. It takes an input vector \(\vec{v}\) and maps it to an output vector \(\vec{v'}\) using the rule: \(A\vec{v} = \vec{v'}\).

1 Mapping the Unit Square

Given the transformation matrix \(M = \begin{pmatrix} 2 & 0 \\ 0 & 1.5 \end{pmatrix}\), transform the four vertices of the square below.

Vertex A: (0, 0)

Vertex B: (1, 0)

Vertex C: (1, 1)

Vertex D: (0, 1)

Plot original and transformed points

Original

2 Reverse Engineering the Transformation

You are given an input vector \(\vec{v} = \begin{pmatrix} 2 \\ 4 \end{pmatrix}\) and an output vector \(\vec{v'} = \begin{pmatrix} -2 \\ 4 \end{pmatrix}\). What is a possible transformation matrix \(A\) that could have caused this?

\(A \begin{pmatrix} 2 \\ 4 \end{pmatrix} = \begin{pmatrix} -2 \\ 4 \end{pmatrix}\)

Show your algebraic reasoning:

Your Matrix \(A\):

\(\begin{pmatrix} \underline{\quad} & \underline{\quad} \\ \underline{\quad} & \underline{\quad} \end{pmatrix}\)

3 Predictive Analysis

What geometric effect would the matrix \(S = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}\) have on a shape? Experiment with a point like (0, 1) and describe the result.

Reference: The Matrix Machine

Think of a matrix as a machine with two input gears and two output pipes.

Standard Basis Vectors

The columns of a transformation matrix tell you exactly where the basis vectors \(\vec{i} = \begin{pmatrix} 1 \\ 0 \end{pmatrix}\) and \(\vec{j} = \begin{pmatrix} 0 \\ 1 \end{pmatrix}\) land.

Matrix \(\begin{pmatrix} a & c \\ b & d \end{pmatrix}\) maps \(\vec{i} \to \begin{pmatrix} a \\ b \end{pmatrix}\) and \(\vec{j} \to \begin{pmatrix} c \\ d \end{pmatrix}\)

Self-Check Questions

Q: Does a 2x2 matrix transformation always keep the origin (0,0) at (0,0)? Why or why not?

Q: If the determinant of a matrix is 0, what happens to the area of a transformed shape?

Q: If you transform a square and it becomes a rectangle twice as wide, what does that tell you about the matrix columns?

Teacher Demo Note

"The Mystery Matrix": Start the class by showing a Desmos or GeoGebra file with a square. Apply a matrix like \(\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}\). Ask students to guess what the numbers in the matrix are doing to the square's corners.

Linear Transformations and Computer Graphics

Matrices as Linear Transformations

Scaling Reflecting and Shearing

Rotation Matrices

Composition of Transformations

Computer Graphics Animation Project

Point Mapping Lab Worksheet

Linear Transformation Intro Slides

Transformation Lab Teacher Guide

Scaling and Shearing Slides

Image Editor Lab Activity

Transformation Key Sheet

Rotation Matrix Slides

Clock Rotation Worksheet

Transformation Cheat Sheet

Composition of Transformations Slides

Order Matters Investigation Worksheet

Composition Answer Key

Retro Engine Project Guide

Animation Logic Sheet

Retro Project Rubric

Retro Project Sample Solution

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