Linear Transformation Simulator
Visualize how a 2x2 matrix transforms vectors and the coordinate grid
Linear Transformation Formula
| Matrix | A = [[a,b],[c,d]] |
| Vector transform | A[x,y] = [ax + by, cx + dy] |
| Determinant | det(A) = ad - bc |
How to Read the Simulator
- The gray grid shows original coordinate directions.
- The blue and green arrows show where the basis vectors move.
- The red arrow shows the selected vector before and after transformation.
- The determinant shows how area changes under the transformation.
Example: matrix [[1, 0.5], [0, 1]] is a shear transformation. It keeps vertical height the same while sliding points horizontally based on their y-value.
Frequently Asked Questions
What is a linear transformation?▼
A linear transformation maps vectors to new vectors while preserving vector addition and scalar multiplication.
How does a 2x2 matrix transform a vector?▼
For matrix [[a,b],[c,d]], vector (x,y) maps to (ax + by, cx + dy).
What does determinant mean visually?▼
The determinant describes area scaling. A negative determinant also indicates orientation reversal.
What transformations can matrices represent?▼
2D matrices can represent scaling, rotation, shear, reflection, projection, and combinations of these transformations.
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