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Basis and Coordinate System Visualizer

Run the Basis and Coordinate System Visualizer Fullscreen

About This MicroSim

This visualization demonstrates that the same geometric vector can have different coordinate representations depending on which basis you use. The left panel shows the standard basis (e₁, e₂) while the right panel shows a custom basis (b₁, b₂) that you can modify.

Learning Objective: Students will interpret how the same point has different coordinate representations in different bases by visualizing standard and custom basis vectors simultaneously.

How to Use

  1. Drag the Vector: In either panel, drag the green vector endpoint to move it
  2. Drag Basis Vectors: In the right panel, drag the endpoints of b₁ (red) and b₂ (blue) to change the custom basis
  3. Use Presets: Click preset buttons to see common basis configurations:
  4. Standard: b₁ = (1, 0), b₂ = (0, 1) - matches the standard basis
  5. Rotated 45°: Basis rotated by 45 degrees
  6. Skewed: Non-orthogonal basis vectors
  7. Stretched: Basis vectors with different lengths
  8. Toggle Options:
  9. Show Grid: Toggle grid lines in both panels
  10. Show Projections: Toggle dashed projection lines

Key Concepts Demonstrated

Coordinate Representation

The same geometric vector v has different coordinates in different bases: - In standard basis: v = (x, y) means v = x·e₁ + y·e₂ - In custom basis: [v]_B = (c₁, c₂) means v = c₁·b₁ + c₂·b₂

The Equation

The coordinates [v]_B satisfy: \(\(\mathbf{v} = c_1\mathbf{b}_1 + c_2\mathbf{b}_2\)\)

Change of Basis

When you change the basis vectors, the grid lines in the right panel change to follow the new basis directions. The coordinates change, but the geometric vector stays the same!

Important Observations

  1. Same Vector, Different Numbers: Moving to a stretched basis makes coordinates smaller
  2. Grid Deformation: The grid follows the basis vectors
  3. Parallel Basis Warning: If b₁ and b₂ become parallel, coordinates become undefined
  4. Verification: The formula c₁·b₁ + c₂·b₂ = v is shown at the bottom

Embedding This MicroSim

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<iframe src="https://dmccreary.github.io/linear-algebra/sims/basis-coordinate-visualizer/main.html"
        height="502px"
        width="100%"
        scrolling="no">
</iframe>

Lesson Plan

Grade Level

Undergraduate introductory linear algebra

Duration

20-25 minutes

Prerequisites

  • Vector basics
  • Linear combinations
  • Understanding of basis vectors

Learning Activities

  1. Exploration (5 min):
  2. Start with "Standard" preset and observe both panels show same coordinates

  3. Move the vector and verify coordinates match

  4. Rotated Basis Investigation (5 min):

  5. Click "Rotated 45°"
  6. Notice how coordinates change but vector stays the same

  7. Find a vector where custom coordinates are simpler than standard

  8. Skewed Basis Exploration (5 min):

  9. Click "Skewed"
  10. Observe how grid lines are no longer perpendicular

  11. Verify the linear combination formula still works

  12. Custom Basis Creation (10 min):

  13. Drag b₁ and b₂ to create your own basis
  14. Find a basis where a specific vector has integer coordinates
  15. Try to make b₁ and b₂ parallel and observe the "undefined" warning

Discussion Questions

  1. Why does the same vector have different coordinates in different bases?
  2. What happens geometrically when you stretch a basis vector?
  3. Why do parallel basis vectors make coordinates undefined?
  4. How would you convert coordinates from one basis to another?

Assessment Ideas

  • Given a vector and a basis, calculate the coordinates by hand and verify
  • Explain why the grid deforms when the basis changes
  • Find a basis that makes a given vector have coordinates (1, 1)

References

  1. 3Blue1Brown - Change of basis - Excellent visual explanation
  2. Khan Academy - Change of basis
  3. Strang, G. (2016). Introduction to Linear Algebra (5th ed.). Section 3.5.
  4. Lay, D. C. (2015). Linear Algebra and Its Applications (5th ed.). Section 4.4.