Basis

Reach into a coordinate plane and drag the two basis vectors to warp the entire grid until your shape lands exactly on the target.

How to play

A source shape sits on a coordinate grid alongside a target outline. Moving the two basis vector handles applies a linear transformation to the whole plane — your goal is to morph the shape onto the target.

• Drag the orange handle (the i vector) and the teal handle (the j vector) to reshape the grid.
• The determinant meter shows the signed area scale of your transformation.
• If the two vectors line up, the determinant drops to zero and the matrix goes singular — the grid flattens to a line and you can't win until you pull it back.
• Match the target outline to clear the level.

Controls

• Desktop: click and drag the two vector handles; use Reset to restore the identity grid and New for a fresh puzzle.
• Phone & tablet: drag the handles with your finger — touch is fully supported. Vectors snap to half-integer steps.

Tips & strategy

• Think of where each unit square corner should land; the vectors are literally the images of (1,0) and (0,1).
• Keep the determinant away from zero — a flat, near-singular grid is a dead end.
• A negative determinant means you've flipped the orientation, which is sometimes exactly what the target needs.

What you'll learn

Basis makes change of basis tangible: every drag is a real 2x2 matrix acting on the plane.

• Skills: linear algebra, change of basis, matrices, vectors
• Best for: advanced high school / college
• For parents & teachers: change of basis in linear algebra.

About Basis

Basis makes the abstract idea of a change of basis tangible: every drag is a real 2x2 matrix acting on the plane. It's an original, free browser game with daily and practice modes built around hands-on linear algebra.

A free, original browser game on PlayPendium — no download, no install. Browse all games →