How does this calculator represent a linear transformation?

This calculator represents a linear transformation by a 2×2 or 3×3 matrix. When you enter a matrix A and a vector v, the tool computes the image A·v and, in 2D, visualises how the transformation acts on the plane.

What do the determinant and trace tell me?

The determinant of a 2×2 or 3×3 matrix is the area or volume scaling factor of the linear transformation and indicates whether orientation is preserved or flipped. The trace is the sum of the diagonal entries and, in many cases, relates to the sum of eigenvalues and the average scaling along principal directions.

Can this linear transformation be inverted?

A linear transformation represented by a square matrix A is invertible if and only if det(A) ≠ 0. The calculator highlights when the determinant is numerically close to zero, which means the transformation squashes the space onto a lower-dimensional subspace and has no inverse.

Why is visualising linear transformations useful?

Visualising how a matrix maps basis vectors and shapes such as the unit square helps build intuition. You can immediately see whether the transformation rotates, shears, scales, or reflects the space, which supports deeper understanding of eigenvalues, eigenvectors and change of basis.

Linear Transformation Calculator

Q: What is a linear transformation?

A linear transformation is a function between vector spaces that preserves vector addition and scalar multiplication. In coordinates, a linear transformation on R^n can be represented by multiplying vectors by an n×n matrix.

Matrix · Vector · Visualisation

Explore linear transformations in 2D and 3D. Enter a 2×2 or 3×3 matrix and a vector, compute the image A·v, determinant and trace, and see an interactive visualisation of how the matrix transforms the plane in 2D.

matrix–vector product determinant & trace 2D geometry visualisation linear algebra intuition

Linear transformation inputs (matrix & vector)

Dimension

2D mode comes with an interactive plot; 3D mode focuses on numeric output.

2D presets (A)

Vector presets (v)

2×2 matrix A (2D)

Rows correspond to outputs of basis vectors e₁, e₂.

a₁₁

a₁₂

a₂₁

a₂₂

Vector v in ℝ²

The linear transformation sends v to A·v.

v₁

v₂

In the plot below you will see v and A·v, plus the image of the unit square.

2D visualisation of the linear transformation

Works in 2D mode only

The canvas shows the original axes, the unit square, the transformed unit square A(Q), and the vector v together with its image A·v.

Original axes & unit square

Transformed basis vectors A e₁, A e₂

Vector v

Transformed vector A·v

Try different matrices (shears, scalings, rotations) and vectors to see how linear transformations stretch, rotate and reflect the plane while keeping grid lines straight.

What is a linear transformation?

Informally, a linear transformation is a map that sends vectors to vectors while preserving the structure of vector addition and scalar multiplication. A function \(T: \mathbb{R}^n \to \mathbb{R}^m\) is linear if for all vectors \(\mathbf{u}, \mathbf{v}\) and scalars \(c\),

\[ T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v}),\quad T(c\,\mathbf{u}) = c\,T(\mathbf{u}). \]

In coordinates, any linear transformation can be represented as multiplication by a matrix. When you enter a matrix \(A\) and a vector \(\mathbf{v}\) in this calculator, the result \(A\mathbf{v}\) is exactly the image \(T(\mathbf{v})\).

Matrix representation in 2D

A 2D linear transformation \(T: \mathbb{R}^2 \to \mathbb{R}^2\) is represented by a 2×2 matrix \[ A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}. \] For a vector \(\mathbf{v} = (x, y)\), \[ A\mathbf{v} = \begin{bmatrix} a_{11}x + a_{12}y \\ a_{21}x + a_{22}y \end{bmatrix}. \]

Geometrically, the columns of \(A\) are the images of the basis vectors \(e_1 = (1, 0)\) and \(e_2 = (0, 1)\).

Determinant, trace and geometric meaning

For a 2×2 or 3×3 matrix, the determinant and trace give quick insight into the geometry of the linear transformation:

Determinant \(\det(A)\): area (2D) or volume (3D) scaling factor. If \(\det(A) = 0\), the transformation collapses space onto a lower-dimensional subspace (no inverse).
Sign of determinant: positive means orientation is preserved; negative means the transformation includes a reflection.
Trace \(\mathrm{tr}(A)\): sum of diagonal entries. In many contexts, it equals the sum of eigenvalues and relates to the average scaling effect along principal directions.

Typical types of linear transformations

Pure scaling: multiplies every vector by the same factor (e.g. diagonal matrices with equal diagonal entries).
Rotation: preserves lengths and angles, with determinant \(+1\).
Reflection: flips orientation (determinant negative) while preserving some distances.
Shear: slides points along a direction while keeping one axis fixed, changing area but keeping some lines parallel.
Projection: collapses vectors onto a line or plane, always with determinant 0.

Using this calculator for learning and practice

This tool is useful for students of linear algebra, data scientists and engineers who need intuition about matrices. You can:

Test examples from textbooks by entering the matrix and vectors directly.
Explore how the determinant changes when you adjust individual matrix entries.
Visualise how shears, rotations and reflections act on the unit square.
Relate geometric effects to concepts like eigenvalues, eigenvectors and rank.