Transforming Space

Linear Algebra (Matrices) · Axiom Academy

A 2×2 matrix is a machine that grabs the whole plane and warps it — rotating, stretching, shearing, or flipping every point at once. A matrix doesn't just sit there — it moves the plane In a first algebra course a matrix is a silent grid of numbers. The secret of linear algebra is that those four numbers are really a set of instructions for transforming space : feed in any point, and the matrix sends it somewhere new. Do that to every point and the entire grid warps as one. Watch the rotation matrix go to work. The plain unit grid — with the basis arrows pointing right and pointing up — is carried a quarter-turn to a brand-new grid. The two arrows land exactly on the matrix's two columns , and a sample vector rides along to its image. The new grid is just the old grid with and sent to where the matrix says — everything else follows along in straight lines. Four ways to warp the same plane Tap a transformation and watch the grid re-warp instantly. Each one is a different 2×2 matrix, and the live matrix shows you its two columns — the landing spots of and . The green sample vector v is dragged to its image M · v every time. Scale fattens areas, shear slants them, flip turns them inside out (det < 0) — all from where two arrows land.

This is the written version of the interactive lesson above. See the full Linear Algebra (Matrices) course.