Linear maps — the series
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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.
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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.
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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.
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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.
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Two graphs the eigenvalues cannot tell apart
A graph's matrix has eigenvalues, and they count a surprising amount of the drawing: its edges, its triangles, every closed walk of every length. They do not count everything. A star with four arms and a square beside a lone point have the same eigenvalues exactly, although one of them is in two pieces — and on six points ten of the 156 graphs have a twin of this kind.
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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.
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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.