Watch the entire coordinate grid warp and stretch. See how matrices transform all of space at once.
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.
Select different transformations and watch the entire coordinate grid respond. Notice how 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.
Different matrices produce different effects. Compare rotations, scalings, shears, reflections, and projections side by side.
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 matrix entries directly and see instant results. Experiment! What happens when you make the determinant zero?
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?
Everything extends to three dimensions. A 3×3 matrix transforms space by specifying 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.
Next: What happens when we apply one transformation after another?