Matrix Transformations
The idea that unlocks all of linear algebra: a matrix is not a table of numbers, it's an instruction for bending space. Every point in the plane moves — but in a way so orderly that just two arrows tell you everything. First-year university.
The columns are where the basis lands
Start with the two unit arrows: î pointing right, ĵ pointing up. A 2×2 matrix does one thing — it says where those two arrows should go. Its first column is the new î, its second column is the new ĵ:
Everything else follows for free. Because the transformation is linear — grid lines stay straight, parallel and evenly spaced — once you've pinned down î and ĵ, every other point is just a combination of the two. Move the two arrows and the entire plane comes along.
Bend the plane 🟠🔵
Drag the orange î and blue ĵ tips (they snap to half-units), or use the sliders and preset buttons. Watch the green unit square — its area is the determinant.
The determinant is an area
The green box is the image of the unit square. Its area is exactly |det M| = |ad − bc| — the factor by which the transformation scales every area in the plane. That one number carries a lot:
Zero → collapse
When det = 0 the box flattens to a line: the whole plane gets squashed onto a lower dimension. Information is lost, and the matrix has no inverse — you can't undo a squash. Try the "Squash" preset.
Negative → flipped
A negative determinant means the transformation turned the plane over, like a page — orientation reversed. î and ĵ have swapped their clockwise/counter-clockwise order. Try "Reflect".
So det = 2 doubles areas, det = 1 (rotations, shears) preserves them, det = 0.5 halves them, and a negative det does all that and mirrors. It's the single most useful number a matrix carries.