Vector Space Span

A Vector Space Span is a Vector Space that is generated by two or more sets of vectors.

• AKA: Vector Span, Span.
• Example(s):
  • $Span(v_1,v_2)=\{rv_1+sv_2:r,s \in R\}$
• Counter-Example(s):
  • Geometric Span,
  • Set Span,
  • Link Span.
• See: Vector Space, Vector Basis, Linear Independence, Linear Combination, Null Space, Vector Space.

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References

2020

• (Weisstein, 2020) ⇒ Weisstein, Eric W. “Vector Space Span." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/VectorSpaceSpan.html
  • QUOTE: The span of subspace generated by vectors $v_1$ and $v_2$ in $V$ is
    $Span(v_1,v_2)=\{rv_1+sv_2:r,s \in R\}$.

    A set of vectors $m=\{v_1,...,v_n\}$ can be tested to see if they span n-dimensional space using the following Wolfram Language function:

    SpanningVectorsQ[m_List?MatrixQ] := (NullSpace[m] == {})

2015

• http://ltcconline.net/greenl/courses/203/Vectors/basisDimension.htm
  • Basis: In our previous discussion, we introduced the concepts of span and linear independence. In a way a set of vectors S = {v1, ..., vk} span a vector space V if there are enough of the right vectors in S, while they are linearly independent if there are no redundancies. We now combine the two concepts.
  • Definition of Basis: Let V be a vector space and S = {v1, v2, ..., vk} be a subset of V. Then S is a basis for V if the following two statements are true.
    • 1. S spans V.
    • 2. S is a linearly independent set of vectors in V.
  • We have seen that any vector space that contains at least two vectors contains infinitely many. It is uninteresting to ask how many vectors there are in a vector space. However there is still a way to measure the size of a vector space. For example, R3 should be larger than R2. We call this size the dimension of the vector space and define it as the number of vectors that are needed to form a basis. Tow show that the dimensions is well defined, we need the following theorem.

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