Arrow keys move the tip of:
Grid:
Blue grid lines are the images of the vertical lines of the input grid, amber ones the images of the horizontal lines. The curved arrow inside the square turns the other way when the picture is mirrored.
Mode
The matrix T
Column 1 is where e₁ lands, column 2 is where e₂ lands.
Pick the presets to chain
How many steps:
Presets
Pick a preset, or edit the numbers.
det T
area of the image of the unit square = |det T|
Area of the image
Te₁ (column 1)
Te₂ (column 2)
Corners: S and T @ S
Step by step
Each picture shows where the unit square and the grid are after that step. The last one is the whole chain at once.
The product, read right to left
Determinants multiply
Does the order matter?
What to try
- Press Identity, then drag the tip of Te₂ around. Watch the blue lines tilt while the amber lines stay put.
- Press Reflect: det T is −1, the area is unchanged, and the curved arrow turns the other way.
- Press Project / squash: the whole grid collapses onto a line, det T = 0, and no matrix can undo that.
- Switch to Combine with rotate then shear, then press Swap the order. The dashed purple outline is where the square lands in the other order.
- In every combination, det(product) equals the product of the dets, even when the matrices themselves do not agree.