Concept

Linear map

A function between vector spaces that respects addition and scaling, sending lines through the origin to lines or to the origin. Once bases are chosen it is a matrix, and its kernel and image measure what it loses and what it reaches.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

Two coefficient arrays, one with determinant zero and one without. The Sylvester matrices of two pairs of polynomials drawn as grids of coefficients, one pair sharing a root and one not, with each determinant computed in whole numbers and checked against whether a shared root exists.

A shared root, found without finding it

Two polynomials have a root in common exactly when one determinant built from their coefficients is zero. No root is computed, nothing is approximated, and the same construction turns two equations in two unknowns into one equation in one.

algebra · Polynomial roots

Every magic square of order three is a point in a space of three. Basis J, U = (1,-1,0,-1,0,1,0,1,-1), V = (0,-1,1,1,0,-1,-1,1,0); Lo Shu = 5J − 3U + V; dimension of the order-3 magic space 3.

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.

algebra · Linear maps

Named alongside it

The objects these essays reach for when they reach for this one.

Determinant Coefficient Curve Dimension Discriminant Elimination Exhaustive search Involution Kernel Magic square Polynomial Rank

All concepts