Skip to content

Linear Transformation Fundamentals Visualizer

Run the Linear Transformation Visualizer Fullscreen

Edit the MicroSim with the p5.js editor

About This MicroSim

This interactive visualization demonstrates the fundamental concepts of linear transformations:

  1. Basis vectors determine everything: The transformation is completely defined by where the standard basis vectors e₁ = (1,0) and e₂ = (0,1) map to
  2. Grid structure is preserved: Linear transformations map parallel lines to parallel lines (or points)
  3. The matrix columns are the transformed basis vectors: The transformation matrix A has columns T(e₁) and T(e₂)

How to Use

  • Drag the handle points on T(e₁) and T(e₂) in the transformed space to define any linear transformation
  • Use the Morph slider to animate between the identity transformation and your current transformation
  • Select presets from the dropdown to see common transformations (rotation, scaling, shear, reflection)
  • Toggle checkboxes to show/hide the grid and sample vector

Embedding

You can include this MicroSim on your website using the following iframe:

1
<iframe src="https://dmccreary.github.io/linear-algebra/sims/linear-transform-basics/main.html" height="482px" scrolling="no"></iframe>

Lesson Plan

Learning Objectives

By using this simulation, students will be able to:

  1. Explain how the columns of a transformation matrix relate to the images of basis vectors
  2. Predict how a linear transformation will affect arbitrary vectors based on its matrix
  3. Identify common transformations (rotation, scaling, shear, reflection) by their matrix form

Suggested Activities

  1. Discovery: Start with identity, then drag T(e₁) and T(e₂) to see how the grid deforms
  2. Prediction: Given a transformation matrix, predict where a sample vector will map before checking
  3. Recognition: Use presets to learn the characteristic matrix patterns for rotation, scaling, shear, and reflection

Assessment Questions

  1. If T(e₁) = (2, 0) and T(e₂) = (0, 2), what type of transformation is this?
  2. What matrix represents a 90° counterclockwise rotation?
  3. Why does a shear transformation preserve area but not angles?

References