Linear Transformation Calculator
Apply a matrix transformation to a vector and visualise the mapped coordinates.
About Linear Transformation Calculator
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)
Worked example
With the default values below, Linear Transformation Calculator returns A·x (transformed vector) of [-4, 3]. Change any input above and every figure here is recalculated in your browser.
Inputs used
- Transformation Matrix A
- 0 -1 1 0
- Input Vector x
- 3, 4
Results
- A·x (transformed vector)
- [-4, 3]
- ||x||
- 5.000000
- ||Ax||
- 5.000000
Step by step
- 1Matrix A= 0 -1 1 0
- 2Input x= [3, 4]
- 3Output A·x= [-4, 3]
- 4||x||= 5
- 5||A·x||= 5
- 6Scaling factor ||Ax||/||x||= 1
Assumptions and limits
- The result is derived only from the 2 inputs listed above. Anything not among them is not part of the calculation.
- In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
- Figures are rounded for display to 6 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.
How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.
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.