Watch the entire coordinate grid warp and stretch. See how matrices transform all of space at once.
A matrix isn't just a grid of numbers—it's an action that moves the whole plane at once. Its columns say where the basis vectors land, and that decides everything.
Pick a transformation and watch the whole coordinate grid move. Notice that grid lines stay straight and parallel lines stay parallel.
A matrix transformation is completely determined by where it sends the basis vectors î and ĵ. Watch how the entire grid follows along.
Here's the key insight: a matrix isn't just a grid of numbers—it's an action. Every matrix takes the entire coordinate grid and transforms it in one smooth motion: stretching, rotating, shearing, reflecting.
The columns of a matrix tell you exactly where the basis vectors land. Once you know that, you know everything about the transformation.
A 2×2 matrix M transforms a vector v by sending it to Mv. The first column is where î lands, the second column is where ĵ lands. That's it—the entire transformation is encoded in just four numbers.
Compare rotations, scalings, shears, reflections, and projections side by side. Find the one that flattens the plane onto a line and loses information.
Each 2×2 matrix produces a different transformation. The yellow square shows how the unit square transforms. Notice how some operations preserve area (rotations) while others change it (scaling).
Edit the four matrix entries or start from a preset, and watch the grid respond. Make the second column match the first: the determinant hits zero and the plane collapses.
The first column is where î (red) lands. The second column is where ĵ (green) lands. Try making the determinant zero—what happens to the grid?
Choose a preset and watch 3D space move. A 3×3 matrix just says where î, ĵ, and k̂ land.
The amber cube shows the transformed unit cube. Watch how the three basis vectors (red î, green ĵ, blue k̂) move—they completely determine the transformation.
Spot a mistake or have an idea?