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Affine, conical, and convex combinations

Main article: Linear span

Take an arbitrary field K, an arbitrary vector space V, and let v1,…,vn 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 ={v1,…,vn}. 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\}.\,}

\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 v1,…,vn, 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}).}

v=\sum a_{i}v_{i}=\sum b_{i}v_{i}{\text{ where }}(a_{i})\neq (b_{i}).
c_{i}:=a_{i}-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}.}

0=\sum c_{i}v_{i}.

If that is possible, then v1,…,vn 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 combinationconical combination, and convex combination, and the associated notions of sets closed under these operations.

Type of combinationRestrictions on coefficientsName of setModel space
Linear combinationno restrictionsVector subspace{\displaystyle \mathbf {R} ^{n}}
Affine combination{\displaystyle \sum a_{i}=1}Affine subspaceAffine hyperplane
Conical combination{\displaystyle a_{i}\geq 0}Convex coneQuadrant or Octant
Convex combination{\displaystyle a_{i}\geq 0} and {\displaystyle \sum a_{i}=1}Convex setSimplex