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MATH
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Definitions from Spans

All definitions from Spans.

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Linear combination

Let v1,…,vnv_1,\ldots,v_n be vectors in a vector space VV over FF. A linear combination of these vectors is an expression

λ1v1+⋯+λnvn,λ1,…,λn∈F.\lambda_1v_1+\cdots+\lambda_nv_n, \qquad \lambda_1,\ldots,\lambda_n\in F.

The vectors are fixed while the scalars vary. For non-zero, non-parallel u,v∈R3u,v\in\mathbb R^3, the vectors

λ1u+λ2v,λ1,λ2∈R,\lambda_1u+\lambda_2v,\qquad \lambda_1,\lambda_2\in\mathbb R,

form the plane through the origin determined by uu and vv, with each coefficient moving along one direction.

  1. [1]:Vector space
  2. [2]:Linear independence
  3. [3]:Closure under linear combinations
  4. [4]:Span

Span

For a subset SS (possibly infinite) in a vector space VV over FF, the span of SS is the set of all finite linear combinations of vectors in SS:

span⁡(S)={λ1v1+⋯+λkvk:k≥0, v1,…,vk∈S, λ1,…,λk∈F}.\operatorname{span}(S) = \left\{ \lambda_1v_1+\cdots+\lambda_kv_k: k\geq 0,\ v_1,\ldots,v_k\in S,\ \lambda_1,\ldots,\lambda_k\in F \right\}.

For a finite set S={v1,…,vn}S=\{v_1,\ldots,v_n\}, this is the same as allowing one coefficient for each listed vector. When k=0k=0, the sum is the empty sum 00.

The span is the subspace generated by the chosen vectors. The empty-sum convention gives span⁡(∅)={0}\operatorname{span}(\varnothing)=\{0\}, which is needed for the zero vector space.

  1. [1]:Linear combination
  2. [2]:Basis
  3. [3]:A basis for a generated subspace
  4. [4]:A basis of P1P_1
  5. [5]:Spanning sets count against independent sets
  6. 18 more

Spanning set

A subset SS in a vector space VV is a spanning set for VV when

span⁡(S)=V.\operatorname{span}(S)=V.

A spanning set may contain redundant vectors.

  1. [1]:Span
  2. [2]:span, independence and possible bases
  3. [3]:Seen Questions
  4. [4]:Standard spanning sets
  5. [5]:Three vectors spanning R3\mathbb R^3
  6. 1 more

Column space

If S={v1,…,vn}S=\{v_1,\ldots,v_n\} is a set of vectors in Rm\mathbb R^m and A=[ v1 ⋯ vn ]A=[\,v_1\ \cdots\ v_n\,], then the span of SS is the column space of AA:

col⁡(A)=span⁡(S)={Ac:c∈Rn},\operatorname{col}(A)=\operatorname{span}(S) =\{Ac:c\in\mathbb R^n\},

where cc records the coefficients of the columns. Thus multiplying AA by cc is exactly the linear combination represented by those coefficients.

  1. [1]:Span
  2. [2]:Image
  3. [3]:Test membership in a span

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Linear combinationSpanSpanning setColumn space