This simulator applies a combined scale, shear and rotation matrix to a wireframe cube, holding a fixed gray reference cube alongside the transformed image so you can directly compare basis vectors, signed volume and orientation. Change the operation order and watch a linear transformation stop commuting.
• A real-time 3D view of the transformed solid cube (face colors track the original faces), the fixed gray wireframe reference cube at [−1,1]³ with volume 8, the transformed basis vectors (red eₓ, green eᵧ, blue e_z), and a reference coordinate grid, with home view, focus-selected-part, auto-rotate, expand and toggleable labels. • Seven live controls: x/y/z scale (−2 to 2, negative reflects, zero collapses a dimension), x-shear-from-y (−1.5 to 1.5), rotation about z (−180° to 180°), operation order (scale→shear→rotate or rotate→shear→scale), and a transformation-progress slider (0 to 1, interpolating from the identity matrix). • Play/pause, single-step and larger-step time controls, plus a playback-speed selector from 0.1x to 60x laboratory speed. • Three actions: animate from identity, apply full transformation, and stop. • Six live metrics: signed determinant, transformed cube volume, the lengths of the images of eₓ, eᵧ and e_z, and the dot product of the transformed eₓ and eᵧ (a direct check of whether they're still perpendicular). • A Curves & measurements tab with a basis-vector-lengths chart and a signed-volume/orientation chart, the full model equations, and snapshot measurements. • An Experiments tab with four guided scenarios (volume scaling, pure shear, reflection, collapse and recover), a model-verification bench of independent automated checks, and a timestamped event log with a copyable trial report. • A Learn & assess tab with four guided lessons, a two-question knowledge-check quiz, and a written scope/reference statement linking to a Georgia Tech linear algebra text.
A matrix transformation maps the standard basis vectors to its own columns, and every other vector transforms as a linear combination of those columns. Combining scale (S), shear (H) and rotation (R) into a single matrix A = R H S or A = S H R applies the rightmost operation first under column-vector notation — so scaling-then-rotating and rotating-then-scaling generally land at different final images, which is exactly what the operation-order selector lets you compare directly on the same starting cube.
The determinant of the combined matrix is a signed volume scale factor: since the reference cube has volume 8, the transformed solid's volume is 8|det A|. The sign of the determinant tracks orientation — a negative determinant means the transformation includes a reflection that flips handedness, even though the magnitude of the volume change is unaffected by that sign.
When a scale factor is driven to exactly zero, the matrix becomes singular: its determinant is zero, the cube collapses into a plane, line or point depending on how many dimensions are zeroed out, and no inverse transformation exists to undo it. The equations panel spells out the scale, shear and rotation matrices explicitly, along with v′ = Av and the volume formula 8|det A|.
The four built-in experiments walk through pure diagonal scaling (det A = 1.5, volume 12), a pure shear that preserves volume but destroys perpendicularity between the x and y basis images, a single-axis reflection (det A = −1, volume unchanged, orientation reversed), and an animated collapse toward a zero z-scale that flattens the cube into a plane at full progress. The model-verification bench in the Experiments tab independently checks determinant, volume and orientation relationships against fresh model instances, leaving your current trial untouched.
A determinant of −2 does not mean a negative physical volume — physical volume always uses |det A|, so the transformed volume doubles. The negative sign instead records that the transformation reverses orientation, meaning a right-handed configuration of the basis vectors becomes left-handed, which is exactly what happens during a reflection.
Matrix multiplication is not commutative in general, and column-vector notation applies the rightmost matrix in a product first. Scaling before rotating moves points to different final positions than rotating before scaling, because each operation acts on whatever configuration the previous operation already produced — you can verify this directly by toggling the operation-order control and comparing the resulting cube and basis vectors.
When any scale factor hits zero, the transformation matrix becomes singular — its determinant is zero and it has no inverse. Geometrically, the cube collapses along that direction: zeroing one axis flattens it into a plane, zeroing two axes collapses it to a line, and the "collapse and recover" experiment in this simulator demonstrates exactly this behavior by animating the z-scale toward zero.
This model covers linear transformations about the origin using column vectors, combining diagonal scaling, a single shear term and rotation about the z-axis only. It does not include translation, perspective projection, or shear terms in other directions, and the progress slider interpolates the scale/shear/angle parameters themselves rather than interpolating the matrix entries directly, so negative target scales pass through genuinely singular states along the way.