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Is span a subset in ##\mathbb{R}^{n}##?

  1. Oct 9, 2015 #1
    1. The problem statement, all variables and given/known data
    Consider the vectors ##\vec{v_{1}},\vec{v_{2}},...,\vec{v_{m}}## in ##\mathbb{R}^{n}##. Is span ##(\vec{v_{1}},...,\vec{v_{m}})## necessarily a subspace of ##\mathbb{R}^{n}##? Justify your answer.

    2. Relevant equations


    3. The attempt at a solution
    I understand the three conditions required for a subset to be a subspace (includes zero vector, closed under addition, closed under scalar multiplication), but I am not sure how to go about testing these properties with the span. Any help would be appreciated. Thanks.
     
  2. jcsd
  3. Oct 9, 2015 #2

    Mark44

    Staff: Mentor

    What's another way to write ##span(\vec{v_{1}},...,\vec{v_{m}})##? How do you know whether a given vector is a member of this set?
     
  4. Oct 9, 2015 #3
    You can rewrite span as the image of a matrix, since the image of a matrix is the span of its columns. Since image is a subspace, then does it follow that span is a subspace?
     
  5. Oct 9, 2015 #4

    Mark44

    Staff: Mentor

    There's no need at all to use matrices. How does your book define the term "span"?
     
  6. Oct 9, 2015 #5
    Span: Consider the vectors ##\vec{v_{1}},...,\vec{v_{m}}## in ##\mathbb{R}^{n}##. The set of all linear combinations ##c_{1}\vec{v_{1}}+...+c_{m}\vec{v_{m}}## of the vectors ##\vec{v_{1}},...,\vec{v_{m}}## is called their span:
    ##span(\vec{v_{1}},...,\vec{v_{m}})=\left \{ c_{1}\vec{v_{1}}+...+c_{m}\vec{v_{m}}:c_{1},...,c_{m} \right \}##
     
  7. Oct 9, 2015 #6

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    OK, so, if ##\vec{w}_1## and ##\vec{w}_2## are in the span, is ##\vec{w}_1 + \vec{w}_2## also in the span? If ##c## is a constant, is ##c \, \vec{w}_1## in the span? Is the vector ##\vec{0}## in the span?
     
  8. Oct 9, 2015 #7

    Mark44

    Staff: Mentor

    Presumably, you mean this:
    ##span(\vec{v_{1}},...,\vec{v_{m}})=\left \{ c_{1}\vec{v_{1}}+...+c_{m}\vec{v_{m}}:c_{1},...,c_{m} \in \mathbb{R} \right \}##
     
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