# Affine, conical, and convex combinations

Main article: Linear span

Take an arbitrary field *K*, an arbitrary vector space *V*, and let *v*_{1},…,*v*_{n} be vectors (in *V*). It’s interesting to consider the set of *all* linear combinations of these vectors. This set is called the *linear span* (or just *span*) of the vectors, say S ={*v*_{1},…,*v*_{n}}. We write the span of S as span(S) or sp(S):{\displaystyle \mathrm {Sp} (v_{1},\ldots ,v_{n}):=\{a_{1}v_{1}+\cdots +a_{n}v_{n}:a_{1},\ldots ,a_{n}\in K\}.\,}

## Linear independence[edit]

Main article: Linear independence

For some sets of vectors *v*_{1},…,*v*_{n}, a single vector can be written in two different ways as a linear combination of them:{\displaystyle v=\sum a_{i}v_{i}=\sum b_{i}v_{i}{\text{ where }}(a_{i})\neq (b_{i}).}

Equivalently, by subtracting these ({\displaystyle c_{i}:=a_{i}-b_{i}}) a non-trivial combination is zero:{\displaystyle 0=\sum c_{i}v_{i}.}

If that is possible, then *v*_{1},…,*v*_{n} are called *linearly dependent*; otherwise, they are *linearly independent*. Similarly, we can speak of linear dependence or independence of an arbitrary set *S* of vectors.

If *S* is linearly independent and the span of *S* equals *V*, then *S* is a basis for *V*.

## Affine, conical, and convex combinations[edit]

By restricting the coefficients used in linear combinations, one can define the related concepts of affine combination, conical combination, and convex combination, and the associated notions of sets closed under these operations.

Type of combination | Restrictions on coefficients | Name of set | Model space |
---|---|---|---|

Linear combination | no restrictions | Vector subspace | {\displaystyle \mathbf {R} ^{n}} |

Affine combination | {\displaystyle \sum a_{i}=1} | Affine subspace | Affine hyperplane |

Conical combination | {\displaystyle a_{i}\geq 0} | Convex cone | Quadrant or Octant |

Convex combination | {\displaystyle a_{i}\geq 0} and {\displaystyle \sum a_{i}=1} | Convex set | Simplex |