The whole sheet at once

The gold and ice-blue arrows are the two columns of the matrix A, the places where e₁ and e₂ land. Drag them on the picture and the whole sheet of graph paper moves with them: the grid lines stay straight, each family of lines stays evenly spaced, and the origin stays put. The pale gold parallelogram is where the unit square went, and the ice-blue ellipse is where the unit circle went. The parallelogram's area is the size of ad − bc; the corner also records the sheet's orientation as a sign, working out ad − bc from the four entries and measuring the signed area on the picture. Drag the second arrow onto the line of the first and the determinant falls to zero: the whole sheet is pressed onto a line, and a coral dashed line appears on the original sheet, every point of which lands on the origin. Drag it on past and the determinant turns negative: the sheet comes back turned over, faintly warm in tint, and the little arc from the gold arrow round to the ice-blue one runs the other way. Switch the view to Its own directions and the picture finds, point by point and straight from the definition, the lines that land back on themselves; the coral arrow Av runs along each, and the factor it is stretched by is the eigenvalue, negative when the arrow is reversed. The corner reads λ twice, once solved from the entries of A and once measured along the lines found. Some arrangements give two lines, others just one; put the gold arrow at (0, 1) and the ice-blue one at (−1, 0), a quarter-turn of the sheet, and there are none; when the arrows are the same multiple of e₁ and e₂, every line is one.