Four numbers, and the whole plane turns, stretches or shears at once. The columns are simply where the axes land.
det 1.00The unit square’s area after the move. One matrix, every point obeys the same rule.
What a matrix is
A machine that moves every arrow at once, by one rule: multiply. Its two columns are simply where the axis arrows (1,0) and (0,1) land — read them off the picture. Every other point is a mix of those two, so it follows.
What that buys you
Rotation, stretch, shear, reflection and any combination are each just four numbers, and applying two moves in a row is one more multiplication. Every layer of a neural network is a matrix like this one — in thousands of dimensions instead of two — followed by a bend.
Evidence. The dots are the same Bible word vectors as the Vectors demo (2-D principal coordinates); the grid lines stay straight, parallel and evenly spaced under every matrix because matrix multiplication is linear — the determinant shown is ad − bc, the signed area of the transformed unit square, computed live.
Nine seconds elsewhere
The foundations in learning order, then the series they lead to. Each lands in about nine seconds.