Find a direction a matrix preserves
When does applying a matrix only scale a selected nonzero vector?
Entering midway? Matrix multiplication combines rows with vector coordinates. Parallel vectors share a line through the origin.
Start with a concrete question
Compare stretching, shearing and 90-degree rotation on three candidate directions.
Try the initial inputs, then write a prediction before changing one setting. All calculations run locally on your CPU.
Go directly to the experiment · Review prerequisite mathematics and return here
What do you expect?
Predict before calculating: When does applying a matrix only scale a selected nonzero vector?
Change one thing. Inspect what happens.
- Calculate Av once before viewing the repeated sequence.
- Find a preserved direction for stretch and shear.
- Try all real candidate directions for rotation.
- Explain eigenvalue as a scale and why testing three vectors alone would not prove nonexistence.
Your settingsMechanism
Calculation belowEvidence
Compare outcomes
Repeated linear transformation
Horizontal: x. Vertical: y.
- 1. A applied 0 times (solid line)
- 2. A applied 1 times (dashed line)
- 3. A applied 2 times (dashed line)
Axes use the displayed ranges.
Read plotted values as a table
| Series | x | y |
|---|---|---|
| A applied 0 times | 0 | 0 |
| A applied 0 times | 1 | 0 |
| A applied 1 times | 0 | 0 |
| A applied 1 times | 2 | 0 |
| A applied 2 times | 0 | 0 |
| A applied 2 times | 4 | 0 |
Follow the calculation
- A=[[2,0],[0,1]]; v=(1,0); Av=(2,0).
- Signed area v_x(Av)_y−v_y(Av)_x=0; zero means these nonzero vectors are parallel.
- Av=2v, so the selected vector is an eigenvector with eigenvalue 2.
- After 0 applications: (1,0).
- After 1 applications: (2,0).
- After 2 applications: (4,0).
An eigenvector is a nonzero direction preserved up to scale by a linear map. A real rotation by 90° has no real eigenvector; the test here examines only the selected direction.
Captures keep your inputs, prediction and observation. Results are recalculated from those inputs. Resetting inputs keeps your captures and challenge history.
Explain the mechanism
Find a direction a matrix preserves
Compare stretching, shearing and 90-degree rotation on three candidate directions.
Concepts to recall
Matrix multiplication combines rows with vector coordinates. Parallel vectors share a line through the origin.
Understand the mechanism
For diag(2,1), (1,0) scales by 2 and (0,1) by 1.
A shear preserves the horizontal direction but moves a vertical vector off its original line.
A 90° real rotation makes every nonzero vector perpendicular to itself. A vector cannot be both parallel and perpendicular to itself unless zero, so this map has no real eigenvector. Complex eigenvectors are outside this activity.
Symbols and calculation
Av=λv, with v≠0. λ (lambda) is a scalar eigenvalue; v is a corresponding eigenvector.
Worked example
A=[[2,0],[0,1]]; v=(1,0); Av=(2,0).
Signed area v_x(Av)_y−v_y(Av)_x=0; zero means these nonzero vectors are parallel.
Av=2v, so the selected vector is an eigenvector with eigenvalue 2.
After 0 applications: (1,0).
After 1 applications: (2,0).
After 2 applications: (4,0).
Your investigation
Calculate Av once before viewing the repeated sequence.
Find a preserved direction for stretch and shear.
Try all real candidate directions for rotation.
Explain eigenvalue as a scale and why testing three vectors alone would not prove nonexistence.
Limits and interpretation
- These are bounded teaching models with explicit assumptions, not evidence of a real-world intervention.
Try a fresh case
A diagonal matrix with entries 3 and 2 acts twice on (1,0). What is the final x coordinate?
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Take the investigation further
Choose any extension. These are project prompts for you to carry out; they are not automatically assessed.
School investigation
Explain a fresh small numerical case with a diagram or table and an argument.
College investigation
Implement the method, test a boundary or counterexample, and justify its assumptions.
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Use a small decision scenario, state units and constraints, and test how the recommendation changes.
Keep the evidence, choose your next step
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