Homogeneous System Explorer
Run the Homogeneous Systems MicroSim Fullscreen
Edit the MicroSim with the p5.js editor
About This MicroSim
A homogeneous system has the form \(Ax = 0\) where all right-hand side values are zero. These systems have special properties:
- The trivial solution \(x = 0\) always exists
- Nontrivial solutions may or may not exist
- The set of all solutions forms a subspace (the null space of \(A\))
Key Concepts
When do nontrivial solutions exist?
| Condition | Nontrivial Solutions? |
|---|---|
| Number of variables > number of equations | Always |
| rank(A) < number of variables | Yes |
| rank(A) = number of variables | No (only trivial) |
Null Space Dimension:
\[\text{dim(null space)} = n - \text{rank}(A)\]
where \(n\) is the number of variables (columns of \(A\)).
Visualizing the Null Space
| Null Space Dimension | Geometric Shape |
|---|---|
| 0 | Just the origin (point) |
| 1 | Line through origin |
| 2 | Plane through origin |
| k | k-dimensional subspace through origin |
Lesson Plan
Learning Objectives
After using this MicroSim, students will be able to:
- Identify when homogeneous systems have nontrivial solutions
- Calculate the dimension of the null space from the rank
- Visualize the null space as a geometric subspace
- Understand why the null space always passes through the origin
Suggested Activities
- Compare Examples: Switch between presets and observe how the null space dimension changes
- Verify the Formula: For each example, verify that dim(null space) = n - rank
- Geometric Intuition: Drag to rotate the 3D view and understand the null space shape
- More Variables: Notice that when n > m, nontrivial solutions are guaranteed