Concept

Image — where it appears

The image of a map is the set of everything it actually reaches: all outputs, as the input ranges over its whole domain. For a linear map it is a subspace, and its dimension, the rank, plus the dimension of what is sent to zero equals the dimension of the domain.

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

A whole line arrives at the origin. A linear map whose determinant is zero, drawn before and after. One line of the plane is sent to the origin and the whole plane is sent onto another line; the dimension lost and the dimension kept add to two.

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.

algebra · Linear maps

A loop that is a hole, and the same loop filled in. a hollow triangle: 3 points, 3 edges, 0 triangles; holes in each dimension 1, 1, 0; a filled triangle: 3 points, 3 edges, 1 triangles; holes in each dimension 1, 0, 0.

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.

algebra · Linear maps

Named alongside it

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

Kernel Matrix Rank Basis Boundary Determinant Dimension Euler characteristic Linearity Non-orientable Nullity Projection

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