Kernel
Named by 9 essays across 2 fields — each of them below, with the objects they name alongside it.
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.
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.
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.
A hole is a cycle that bounds nothing
A hollow triangle and a filled one have the same three edges and the same loop round them. In one the loop is the edge of something and in the other it is not, and that difference — a cycle that is not a boundary — is what a hole is. Counting holes is rank and nullity applied twice, and the Euler characteristic is what is left when the two applications cancel.
The boards on which every light goes out
In Lights Out, pressing a light toggles it and its neighbours, and the goal is to turn every light off. Over the field with two elements the puzzle is a system of linear equations, and whether every pattern can be cleared depends only on whether one matrix is invertible. On a 3 × 3 board it is; on 5 × 5 a quarter of patterns can be cleared; on 39 × 39 one in four billion. Which sizes fail is decided by a common factor of two polynomials.
Every light on, from every light off
In Lights Out, pressing a light toggles it and its neighbours. On a five-by-five board only one pattern in four can be reached from all off — and yet all on is always one of them. The same is true on every board and on every network of lights whatever, provided the influence runs both ways. The proof is a single observation about symmetric matrices over a field with two elements, and it fails, as a three-arrow example shows, the moment influence runs one way.
When a press skips its own light
In Lights Out every board can be turned from all off to all on, because each press toggles its own light. Change the rule so a press toggles only the neighbours and the guarantee is gone: 310 of the 3,600 boards up to sixty by sixty cannot be lit. A single comparison of powers of two predicts every one of them, and on general graphs the answer depends on whether the number of vertices is odd or even, for a reason that is one line of algebra.
Half the chance for every silent set
In the lights game where a press toggles only the neighbours, a random network can be lit about 56 times in a hundred when it has an even number of vertices and 44 when odd. The numbers are 0.560562 and 0.439438, they add to one, and they come from a chain: each new vertex raises or lowers the dimension of the silent sets by one, and each dimension halves the chance.
The magic squares are a space
A magic square is a grid whose rows, columns and diagonals share one sum. The conditions are linear, so the squares form a space: three-dimensional for order three, eight-dimensional for order four, n² − 2n in general. Inside the space of order three the eight classical squares are eight points with coordinates ±1 and ±3. Of the 880 squares of order four, 640 are singular matrices, and the search finds the reason: a symmetry of the rows and columns that carries every number to the one it adds to seventeen with.
Named alongside it
The objects these essays reach for when they reach for this one.
RankDimensionFinite fieldLinear systemMatrixParityBasisDeterminantImageLinearityNullityRandom graph