MATH1231 - Vector Spans

The span of a set is the totality of all things we can build (all linear combinations) from the vectors within it.

For example, the x axis is spanned by the vector (1 0), as anything you multiply it by will leave you with (λ 0).

Correspondingly the y axis is spanned by the vector (0, 1), and any vector (a, b) can be built from those two.

Example

Span of (1 2 0), (3 -2 -1) in R3
this = λ(1 2 0) + μ(3 -2 -1)
this = a plane through the origin and those two points.

Proof that Vector Spans are Subspaces

V = vectorspace, A = (non-empty) set of vectors, show that span(A) is a subspace

Elements of span = v1λ1 + v2λ2 + … + vnλn (vn from A,λn from R)

Existence of a Zero: Let all λ = 0, therefore 0 vector

Closure under Multiplication:
Multiply all by α: αv1λ1 + αv2λ2 + … + αvnλn
Given αλn is a constant in R, this is in a linear combination of A and hence in the span of A

Closure under Addition:
Linear Combination: v1μ1 + v2μ2 + … + vnμn
Linear Combination: v1λ1 + v2λ2 + … + vnλn
Add the two together: (λ1 + μ1)v1 + (λ2 + μ2)v2 + … + (λn + μn)vn

Linear combination and hence belongs to the span.

Having passed the three tests, this is a proven subspace.

A set cannot be a spanning set for Rn if n > dim(span)