Linear transformation: T(v) = Av. A 2×2 matrix maps R² → R². det(A) = area scaling factor. Eigenvalues reveal stretching along invariant directions. Special cases: rotations (det=1, no real eigenvalues), reflections (det=−1), projections (det=0).
How to Use
Define the transformation:
- Matrix: 2×2 entries or preset
- Vector: Input v
- Output: T(v) = Av
Types
- Rotation: [[cos,-sin],[sin,cos]]
- Shear: [[1,k],[0,1]]
- Reflection: det = -1
- Scaling: diagonal
Properties
T(αu+βv) = αT(u)+βT(v). Preserves origin, lines, parallelism. det(A) = area scaling. |det(A)| < 1 contracts, > 1 expands.
Step-by-Step Instructions
- 1Select a transformation.
- 2Enter input vector.
- 3View T(v) = Av.
- 4Check eigenvalues.
- 5Analyze determinant.