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Ladder

Linear maps — the ladder

2 distinct arguments against one idea, from the one that introduces it to the one that assumes the rest.
  1. A linear map redrawing the plane. The integer grid before and after a linear transformation; the shaded unit square becomes a parallelogram whose area is the determinant.

    A matrix is a picture of what happens to the grid

    Four numbers in a box is not an object anyone has intuitions about. The same four numbers, shown as an instruction for redrawing the plane, are.

    rung 1 · algebra
  2. A whole line arrives at the origin. A linear map whose determinant is zero, drawn before and after. One line of the plane is sent to the origin and the whole plane is sent onto another line; the dimension lost and the dimension kept add to two.

    What a map throws away

    A linear map redraws the grid, and the determinant measures how much it stretches area. When that measurement comes out zero the map has flattened the plane onto a line — and the question worth asking is not how much was lost but how much survived, because the two always add to what there was.

    rung 2 · algebra

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