Change of Basis Interactive Visualizer

Run the Change of Basis Visualizer Fullscreen

Edit the MicroSim with the p5.js editor

About This MicroSim

This visualization demonstrates a fundamental concept: the same vector has different coordinate representations in different bases.

Key ideas:

• The vector itself (green arrow) doesn't change—it's the same geometric object
• Only its coordinate representation changes based on which basis we use
• The transition matrix P⁻¹ converts coordinates: [v]_B = P⁻¹[v]_std

How to Use

1. Drag the vector v (green) to see how its coordinates change in both bases
2. Select a preset basis (Rotated, Skewed, Scaled) or use Custom
3. Adjust the rotation slider to rotate the custom basis
4. Toggle "Overlay Bases" to see both coordinate systems on one grid
5. Toggle "Show Grids" to see the coordinate grid lines

Key Observations

• The vector arrow stays in the same position regardless of basis choice
• In a rotated basis, the coordinates reflect the new orientation
• The transition matrix P⁻¹ relates the two coordinate systems

Embedding

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

Lesson Plan

Learning Objectives

Students will be able to:

1. Explain why coordinate representations depend on basis choice
2. Calculate coordinates in a new basis using the transition matrix
3. Distinguish between a vector and its coordinate representation

Suggested Activities

1. Predict coordinates: Before dragging, predict what [v]_B will be
2. Verify transition: Check that P⁻¹[v]_std = [v]_B
3. Special vectors: Find vectors that have the same coordinates in both bases

Assessment Questions

1. If the basis is rotated 90°, how do the coordinates of (1, 0) change?
2. What properties do similar matrices share?
3. Why is change of basis useful in linear algebra?

References

• Chapter 4: Linear Transformations - Change of Basis section
• Similar matrices and diagonalization