widget one · section A, live
The columns are where the basis vectors land
Drag either arrowhead. The purple arrow is column one — it is where (1, 0) ends up. The orange arrow is column two — where (0, 1) ends up. You are not editing a picture and hoping the matrix follows; you are editing the matrix, because the arrowheads are the matrix's four numbers.
the plane, before and after
the same thing, as numbers
Things worth doing on purpose: drag column two until it lies on top of column one and watch the whole plane fall onto a line — that is singular, and the determinant readout hits zero at the same instant. Drag column one past column two so the arrows swap sides and watch the determinant go negative — that is orientation flipping, and the square is now inside-out.