Linear algebra · terminology

Linear, Affine, Conical, Convex

Four ways to combine vectors \(a\mathbf{u} + b\mathbf{v}\): the only difference is which coefficients you allow.

Linear, affine, conical, convex combinations

A combination of vectors is \( a_1\mathbf{x}_1 + a_2\mathbf{x}_2 + \dots + a_n\mathbf{x}_n \).

Type of combinationRestriction on coefficientsName of setModel space
Linearno restrictionsVector subspace\( \mathbb{R}^n \)
Affine\( \sum a_i = 1 \)Affine subspaceAffine hyperplane
Conical\( a_i \ge 0 \)Convex coneQuadrant, octant, or orthant
Convex\( a_i \ge 0 \) and \( \sum a_i = 1 \)Convex setSimplex

Wikipedia: affine, conical and convex combinations

What can two vectors reach?

u v

The same four pictures, by hand

Hand-drawn sketches: (a) linear combinations fill the plane, (b) affine combinations form the line through u and v, (c) conic combinations fill the wedge between u and v, (d) convex combinations form the segment from u to v
source

The same words for maps

Linear map \( f(\mathbf{x}) = A\mathbf{x} \)
Grid lines stay straight and evenly spaced; the origin stays put.

Affine map \( f(\mathbf{x}) = A\mathbf{x} + \mathbf{b} \)
A linear map followed by a shift: the origin moves.

Affine maps keep affine combinations: the midpoint of a segment still lands on the midpoint.