Linear Algebra Visualizer — 3D

The 2D original transforms a flat grid with a 2×2 matrix. This companion applies a real 3×3 matrix to a lattice of points, the three basis vectors and a unit cube in genuine 3D space — the exact same linear-map math, one dimension further.

det(A)

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trace(A)

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Volume scale

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Orientation

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a11 1.0

a12 0.0

a13 0.0

a21 0.0

a22 1.0

a23 0.0

a31 0.0

a32 0.0

a33 1.0

Lattice density5³

Reading the scene — thin blue lines run from every original lattice point to its image A·p, exactly like the 2D original's transformed grid lines, now spanning all three axes. The red/green/blue arrows are the transformed basis vectors A·i, A·j, A·k; the yellow wireframe is the transformed unit cube, whose volume equals |det(A)|.

A·i (was x-axis)

A·j (was y-axis)

A·k (was z-axis)

Unit cube image — volume = |det(A)|

Matrix Transformations — Extended to Real 3D

The 2D original applies a 2×2 matrix to a lattice of points, drawing a line from each original point to its image so you can watch the plane stretch, rotate and shear. This companion ports the exact same idea one dimension further: a 3×3 matrix A is applied to a cubic lattice of points p=(x,y,z), and a line is drawn from every p to A·p, using the identical row-by-row multiplication (x'=a11x+a12y+a13z, and so on) as the flat version's x'=a11x+a12y.

Three colored arrows track where the basis vectors i, j and k land after the transform — the 3D generalization of the "how does the unit square deform" question the 2D grid answers implicitly. A yellow wireframe cube tracks the image of the unit cube [0,1]³; its enclosed volume equals |det(A)|, computed live via the standard 3×3 cofactor expansion. The trace and determinant readouts update on every slider drag, and the orientation indicator flips from "Preserved" to "Flipped" exactly when det(A) changes sign — the same invariant that governs whether a 2D transform mirrors the plane. Drag to orbit the camera; scroll to zoom.