Different Coordinate Systems

Linear Algebra (Matrices) · Axiom Academy

An arrow in the plane never moves — but the numbers you use to name it depend entirely on the grid you measure against. The arrow stays put. The numbers don t. A vector is a geometric thing — an arrow with a length and a direction, sitting in the plane. The pair of numbers we usually write for it, like (3, 2) , is not the arrow itself. It is a set of instructions: go 3 along one ruler, then 2 along another. Change the rulers — the basis — and the same arrow gets a different pair of numbers. Watch one arrow hold perfectly still while the grid underneath it tilts from the standard square grid into a skewed one. The arrow does not budge, but its coordinate readout slides to brand-new values: those are its coordinates in the new basis . Same arrow, different rulers, different numbers — neither reading is more correct than the other. Here the orange point is pinned in place — it is one fixed location in the plane. Drag the blue and green basis arrows to set up your own coordinate system. The point never moves, but its coordinates — how many of your b_1 and b_2 it takes to reach it — are recomputed the instant you let go of either arrow. Tilt the rulers and the same point reads differently; line them back up with the axes and the two readings agree again.

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