Linear Transformation

Visualize how a 2×2 matrix transforms the plane

Transformation matrix
Probe vector
[2 1; 1 2] · (1, 1) = (3, 3)
Determinant (area scale)
3
Trace
4
Rank
2
Invertible
Yes
Orientation
Preserved (det > 0)
Where the basis lands
î (1, 0) → (2, 1) · ĵ (0, 1) → (1, 2)
Probe vector image
(3, 3)
Eigenvalues & eigenvectors
Two real eigenvalues — the plane stretches along two independent directions.
  • Eigenvalue λ = 3 → Eigenvector (0.7071, 0.7071)
  • Eigenvalue λ = 1 → Eigenvector (0.7071, -0.7071)
Inverse matrix
[0.6667 -0.3333; -0.3333 0.6667]
Linear TransformationGrid showing the unit square (dashed) mapped to a parallelogram by the matrix, with the transformed basis vectors î and ĵ and the probe vector and its image.xyîĵ