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Determinant — the series

7 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. The unit square, mapped: area × 5. The unit square and the parallelogram it becomes under a linear map, with the area of that parallelogram computed from its own corners and set against ad − bc.

    The number that says how much room is left

    A linear map takes the unit square to a parallelogram. The area of that parallelogram is one number, it is computable from the four entries of the matrix, and almost everything the determinant is used for is a restatement of that sentence.

    part 1 · algebra
  2. A parallelepiped of volume 2.94. The image of the unit cube under a three-by-three matrix, beside the six signed products whose sum is its volume.

    The only function that behaves like a volume

    Ask for a function of the columns of a matrix that scales when a column scales, vanishes when two columns agree, and gives one on the identity. Three conditions, and there is exactly one such function in every dimension.

    part 2 · algebra
  3. A lattice of determinant 3, and the ellipse that must hold a point. A lattice with the parallelogram its basis spans, an ellipse centred at the origin, and the nearest non-zero lattice point it contains.

    One point in every big enough shape

    A determinant measures a lattice, not the basis that happened to describe it — and that measurement is an exchange rate. Any symmetric convex region with more than four times that area has to swallow a lattice point.

    part 3 · algebra
  4. A determinant counting the 16 spanning trees. A small graph, the minor of its Laplacian, and every one of its spanning trees drawn as thumbnails.

    A determinant that counts trees

    Write down a graph's Laplacian, strike out one row and its column, take the determinant. The answer is the number of spanning trees — and the minus signs in the determinant are what cancel every subset of edges that is not one.

    part 4 · algebra
  5. A determinant of −5 and a permanent of 23 from the same six products. The six products of a three-by-three matrix listed once, added with signs to give the determinant and without signs to give the permanent, with a row operation applied to both.

    The same sum without its minus signs

    Delete the signs from the determinant's sum over permutations and what is left counts things directly rather than by cancellation. It is a better count and a far worse object — because the cancellation was what made the determinant computable.

    part 5 · algebra
  6. The same two lengths at 3 angles, and the area largest at the right angle. Parallelograms spanned by columns of lengths 1.6 and 1.25 at angles 38, 90, 142 degrees. Their areas are 1.23, 2.00, 1.23; the bound 2.00 is the product of the lengths and is reached only when the columns are perpendicular.

    The biggest box built from signs

    Fill a square table with plus and minus ones and ask how large its determinant can be. The columns all have the same length, so the answer is a box with fixed edges — largest when every corner is square, which is possible only when the size is a multiple of four.

    part 6 · algebra
  7. A map that folds the plane over itself, with every point still counted once. The map (u, v³ + uv): its domain shaded by the sign of the Jacobian determinant and its image with the grid carried across. At 5 marked target points the preimages number 3, 3, 1, 1, 1 and their signed counts are all 1; the determinant integrates to 4.447, equal to the integral of the signed count.

    The count a fold cannot change

    A curved map can fold the plane over itself, so that one point has three preimages and its neighbour has one. Count each preimage with the sign of the determinant there and the jump disappears — the signed count is the same everywhere, and it is a whole number.

    part 7 · algebra

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