Linear Transformation Calculator
Apply a matrix transformation to a vector and visualise the mapped coordinates.
Inputs
A·x (transformed vector)
[-4, 3]
||x||
5.000000
||Ax||
5.000000
Step by step
Matrix A
= 0 -1 1 0
Input x
= [3, 4]
Output A·x
= [-4, 3]
||x||
= 5
||A·x||
= 5
Scaling factor ||Ax||/||x||
= 1
How it works
A linear transformation T(x) = Ax maps a vector x through the matrix A. Each output component is the dot product of the corresponding row of A with x. Common 2D examples include rotations, reflections, and scaling. The matrix A provided here (default) is a 90° counter-clockwise rotation.
Formula
Linear Map
(Ax)ᵢ = Σⱼ Aᵢⱼ xⱼ
- A
- Transformation matrix (m×n)
- x
- Input vector (n×1)
Frequently Asked Questions
What transformations can a matrix represent?
Any linear transformation: rotations, reflections, scaling, shearing, projections, and compositions of these. Non-linear transformations (like translation) require augmented matrices.
Does this preserve vector length?
Only if A is an orthogonal matrix (AᵀA = I). Orthogonal matrices represent rotations and reflections which preserve lengths and angles.