Linear Transformation Calculator
Linear transformation calculator with 2×2 and 3×3 matrices. Map vectors with A·v, compute determinant and trace, classify the effect, and see an interactive 2D visualisation of how a matrix transforms the plane.
Full original guide (expanded)
Linear Transformation Calculator
Matrix · Vector · VisualisationExplore 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.
Linear transformation inputs (matrix & vector)
2D visualisation of the linear transformation
Works in 2D mode onlyThe canvas shows the original axes, the unit square, the transformed unit square A(Q), and the vector v together with its image 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.
Related algebra & 3D math tools
Formula (LaTeX) + variables + units
T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v}),\quad T(c\,\mathbf{u}) = c\,T(\mathbf{u}).
A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}.
A\mathbf{v} = \begin{bmatrix} a_{11}x + a_{12}y \\ a_{21}x + a_{22}y \end{bmatrix}.
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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 \(e_2 = (0, 1)\).
- No variables provided in audit spec.
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Last code update: 2026-01-19
- Initial audit spec draft generated from HTML extraction (review required).
- Verify formulas match the calculator engine and convert any text-only formulas to LaTeX.
- Confirm sources are authoritative and relevant to the calculator methodology.