Linear Transformation

Visualize how a 2×2 matrix transforms the plane

[2 1; 1 2] · (1, 1) = (3, 3)

Determinant (area scale)

Trace

Rank

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]

About this calculator

The Linear Transformation Calculator analyzes the map of the plane defined by a 2×2 matrix and applies it to a probe vector. It shows where the standard basis vectors land, the determinant as a signed area scale factor, the trace, rank, orientation (preserved, reversed, or collapsed), the inverse matrix when it exists, and the full eigen-decomposition — real distinct, real repeated, or a complex conjugate pair — with unit eigenvectors. Singular matrices are reported with the dimension the plane collapses to.

How to use the Linear Transformation calculator

Enter the values for your problem into the input fields.
Read the result — it updates instantly as you type.
Check the formula and the visual explanation to follow how the answer was found.
Copy the page URL to share the exact calculation.

Common examples

Scaling [[2,0],[0,3]] sends (1, 1) → (2, 3); det = 6, eigenvalues λ = 3 and 2
Rotation [[0,−1],[1,0]] sends (1, 0) → (0, 1); det = 1, complex eigenvalues 0 ± 1i, orientation preserved
Singular [[1,2],[2,4]] → det = 0, rank 1: the plane collapses onto a line
Shear [[1,1],[0,1]] sends (2, 3) → (5, 3); repeated λ = 1, inverse [[1,−1],[0,1]]

Frequently asked questions

How does the Linear Transformation calculator work?

Enter your values and the calculator applies the exact mathematical method for this problem, showing the result together with the formula it used. Everything is computed with high-precision arithmetic, so the answer you see is not limited by ordinary floating-point rounding.

When would I use the Linear Transformation calculator?

It is useful for homework and exam preparation, for checking work done by hand, and whenever a step in a larger problem needs this calculation done quickly and reliably.

How accurate are the results?

Calculations run at 30 significant digits of precision internally. Displayed values are rounded for readability, but the underlying result is far more precise than a typical hand or pocket-calculator computation.

Can I share a specific calculation with someone else?

Yes. Every value you enter updates the URL, so copying the address bar and sharing the link reopens the exact same calculation for whoever clicks it.