All simulators
Matrix transformations
Drag a 2×2 matrix and watch the plane stretch, flip and rotate. Area = determinant.
Drag a 2×2 matrix and watch the plane stretch, flip and rotate. Area = determinant.
Drag the tips of î and ĵ, or the orange point v. Everything else follows.
A 2×2 matrix is a recipe for moving every point of the plane. Its first column says where î = (1, 0) lands and its second column says where ĵ = (0, 1) lands. Any point v = (x, y) is x·î + y·ĵ, so it lands on x·(first column) + y·(second column). Grid lines stay straight, parallel and evenly spaced, and the origin never moves.
The determinant ad − bc is the signed area of the unit square after the move. If it is 2, every shape doubles in area; if it is negative, the plane has been flipped over (î to ĵ turns clockwise instead of anticlockwise); if it is 0, the plane is squashed flat and no inverse matrix can undo it.
AB means “do B first, then A”, because (AB)v = A(Bv). Changing the order usually changes the answer.
A general linear map: straight lines stay straight and the origin stays fixed.
Inverse M⁻¹
Applying M⁻¹ after M moves every point back home.
M × v for the orange point
Same thing, read by columns: Mv = 1·(î’s image) + 2·(ĵ’s image).
Challenges 0/3
A 2×2 matrix is a recipe for moving every point of the plane. Its first column says where î = (1, 0) lands and its second column says where ĵ = (0, 1) lands. Any point v = (x, y) is x·î + y·ĵ, so it lands on x·(first column) + y·(second column). Grid lines stay straight, parallel and evenly spaced, and the origin never moves.
The determinant ad − bc is the signed area of the unit square after the move. If it is 2, every shape doubles in area; if it is negative, the plane has been flipped over (î to ĵ turns clockwise instead of anticlockwise); if it is 0, the plane is squashed flat and no inverse matrix can undo it.
AB means “do B first, then A”, because (AB)v = A(Bv). Changing the order usually changes the answer.
Things to try