Linear Transformation

See a 2×2 matrix as geometry. Drag the entries and watch the plane stretch, rotate, shear, and flip — with eigenvectors and determinant drawn live.

Linear Transformation StudioLive

2×2 matrices as geometry

Controls

Set the matrix [[a, b], [c, d]] and watch how it warps the grid, the unit square, and the unit circle. î and ĵ show where the basis vectors land.

Presets

a (row 1, col 1)

b (row 1, col 2)

c (row 2, col 1)

d (row 2, col 2)

▶ Run in Python

Data Inspector

det (signed area)1.000

trace (a + d)2.000

eigenvaluesreal

λλ₁ = 1.00, λ₂ = 1.00

orientationpreserved

Governing equation

M = [1001], det = a d - b c = 1.00

Eigenvalues

λ = \frac{( a + d ) \pm ( a + d ) ^{2} - 4 ( a d - b c )}{2}

Reading this result: A determinant of 1.00 (between 0 and 1) shrinks areas to 100% of the original while preserving orientation.

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How it works

Every 2×2 matrix [[a, b], [c, d]] is a linear transformation of the plane: it sends the basis vectors î and ĵ to new positions and drags the whole grid with them. This studio maps the unit square to a parallelogram (whose signed area is the determinant), the unit circle to an ellipse, and — when the eigenvalues of λ² − (a+d)λ + (ad−bc) = 0 are real — draws the invariant eigenvector directions the transform only scales. A determinant above 1 expands area, between 0 and 1 shrinks it, below 0 flips orientation, and exactly 0 collapses the plane onto a line.

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Frequently asked questions

Is this 2D linear transformation matrix tool really free?▾

Yes. Linear Transformation runs entirely in your browser using your device's own compute, so local use is free forever. You only pay Compute Tokens if you scale a job to the cloud.

Do I need to install anything?▾

No. Everything runs client-side in a modern browser — no downloads, no license, no account required to start.

Can I save or share my simulation?▾

Create a free account to save projects, and use a shareable embed or minted DOI to publish a live, interactive version anywhere.

How accurate are the results?▾

The solver uses established numerical methods, but results are for research and educational purposes and should be validated against experiment or professional review before you rely on them.