Kernel and Range Interactive Visualizer
Run the Kernel and Range Visualizer Fullscreen
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About This MicroSim
This visualization shows the two fundamental subspaces of a linear transformation:
- Kernel (Null Space): Vectors that map to zero, shown in gray
- Range (Column Space): All possible outputs, shown in red
The Rank-Nullity Theorem states:
\[\text{dim(Domain)} = \text{Rank} + \text{Nullity}\]
For a 2×2 matrix, this means Rank + Nullity = 2.
How to Use
- Click "Full Rank" to generate an invertible matrix (rank 2, nullity 0)
- Click "Rank Deficient" to generate a rank-1 matrix with nontrivial kernel
- Toggle "Show Kernel" to highlight the null space direction
- Toggle "Show Mapping" to see how vectors map from domain to codomain
- Click "Animate" to watch vectors transform
Key Observations
- Full rank: Kernel = {0}, Range = all of ℝ², transformation is invertible
- Rank deficient: Kernel is a line, Range is a line, dimension collapses
Watch how vectors in the kernel (gray) all collapse to the origin, while vectors outside the kernel map to the range subspace.
Embedding
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Lesson Plan
Learning Objectives
Students will be able to:
- Identify the kernel and range of a transformation visually
- Verify the rank-nullity theorem for 2×2 matrices
- Explain why rank-deficient transformations are not invertible
Suggested Activities
- Verify theorem: For each matrix, check that rank + nullity = 2
- Trace vectors: Follow a specific vector from domain to codomain
- Kernel test: Given a vector, predict if it's in the kernel
Assessment Questions
- If a 3×4 matrix has rank 2, what is its nullity?
- Why can't a transformation with nontrivial kernel be inverted?
- How are the columns of a matrix related to its range?
References
- Chapter 4: Linear Transformations - Kernel and Range section
- Rank-Nullity Theorem