What does the determinant mean geometrically?

It is the factor by which the transformation scales area (volume in 3D). The unit square is mapped to the parallelogram spanned by the matrix columns, and its area is exactly |det A|. The sign tells you about orientation: positive means only stretching and turning, negative means the plane has also been flipped over (counter-clockwise becomes clockwise), and zero means everything is squashed onto a line or a point, so the matrix has no inverse.

How do I spot eigenvectors in the picture?

An eigenvector keeps its direction under the transformation and is only scaled by λ (reversed if λ is negative). The tool draws the lines through real eigenvectors as purple dashed lines; during the animation, points on those lines stay on them. A rotation (other than 0° or 180°) has no real eigenvectors — it has a pair of complex conjugate eigenvalues instead, which signals a rotation part.

Why is AB usually different from BA?

Matrix multiplication is composition of transformations, read right to left: AB means do B first, then A. Shearing then rotating usually leaves the grid in a different place from rotating then shearing, so AB ≠ BA. Add the two steps in both orders under Composition and play them to see the difference, along with the combined matrix.

Why can’t a projection be inverted?

A projection flattens the whole plane onto a line, so many different points end up at the same spot. That information is lost and there is no unique way back, so no inverse exists. In numbers, a projection matrix has determinant 0 and rank 1, with eigenvalues 1 (vectors on the line stay put) and 0 (the perpendicular direction is crushed).

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▱Linear Transformation Visualizer - Matrices, Determinants

“A matrix is a linear transformation” clicks once you see it move. In 2D, type a 2×2 matrix (expressions such as cos(pi/6) or sqrt(2)/2 work) or pick a preset — rotation, scaling, horizontal or vertical shear, reflection in an axis or any line, projection onto the x-axis or any line — and fine-tune the angle or factor with a slider. Press play and the whole grid glides from the identity to your matrix; matrices with a rotation part are interpolated by angle, so they turn instead of collapsing through a point. The basis vectors î and ĵ land on the columns of the matrix, and the unit square becomes a parallelogram whose area is |det A|; it changes colour when the determinant is negative and the orientation flips like a mirror image, and a zero determinant warns that the plane has been squashed and cannot be undone. Real eigenvectors are drawn as dashed lines — vectors along them are only stretched by λ — while complex eigenvalues are explained as a rotation. You can also transform a letter F or a smiley, swap in the inverse matrix with one click, and queue several steps to play in order, seeing why “shear then rotate” differs from “rotate then shear” and how the combined matrix multiplies right to left. 3D mode takes a 3×3 matrix: the unit cube turns into a parallelepiped with volume |det A| in a view you can drag around, with eigenvalues, eigenvectors and the inverse listed.

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How to use Linear Transformation Visualizer

1Choose 2D or 3D, type a matrix or click a preset and drag its parameter
2Press Play to watch the grid, basis vectors and shape move from the original to the image
3Read the determinant, orientation, eigenvalues and eigenvectors and the inverse
4Add several steps under Composition, play them in order and read the combined matrix

FAQ

What does the determinant mean geometrically?: It is the factor by which the transformation scales area (volume in 3D). The unit square is mapped to the parallelogram spanned by the matrix columns, and its area is exactly |det A|. The sign tells you about orientation: positive means only stretching and turning, negative means the plane has also been flipped over (counter-clockwise becomes clockwise), and zero means everything is squashed onto a line or a point, so the matrix has no inverse.
How do I spot eigenvectors in the picture?: An eigenvector keeps its direction under the transformation and is only scaled by λ (reversed if λ is negative). The tool draws the lines through real eigenvectors as purple dashed lines; during the animation, points on those lines stay on them. A rotation (other than 0° or 180°) has no real eigenvectors — it has a pair of complex conjugate eigenvalues instead, which signals a rotation part.
Why is AB usually different from BA?: Matrix multiplication is composition of transformations, read right to left: AB means do B first, then A. Shearing then rotating usually leaves the grid in a different place from rotating then shearing, so AB ≠ BA. Add the two steps in both orders under Composition and play them to see the difference, along with the combined matrix.
Why can’t a projection be inverted?: A projection flattens the whole plane onto a line, so many different points end up at the same spot. That information is lost and there is no unique way back, so no inverse exists. In numbers, a projection matrix has determinant 0 and rank 1, with eigenvalues 1 (vectors on the line stay put) and 0 (the perpendicular direction is crushed).