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Linear maps — the series

4 essays on 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.

    part 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.

    part 2 · algebra
  3. 2 independent rows and 2 independent columns. An array of 3 rows and 4 columns beside its transpose, with the independent rows of each shaded, showing the same count on both.

    Counted across and counted down

    A rectangular array has a number of independent rows and a number of independent columns. The two are counted in different spaces, from different objects, by computations that share nothing — and they are always the same number, which is why 'rank' is one word.

    part 3 · algebra
  4. 2 independent cycles and 3 independent cuts, on 5 edges. A small graph beside its incidence matrix, with the matrix's rank and nullity given and shown to be the number of independent cuts and the number of independent cycles.

    The cycles and the cuts

    The count that splits a map's source into what dies and what survives has nothing to do with graphs. Apply it to a matrix built from a graph's edges and points and it says that a graph's independent cycles and its independent cuts add to its number of edges — a theorem about drawings, obtained from an array.

    part 4 · algebra

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