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Do these sets span the same space?

  1. May 4, 2013 #1
    Suppose you have two sets [itex]S_{1}[/itex] and [itex]S_{2}[/itex]. Suppose you also know that every vector in [itex]S_{1}[/itex] is expressible as a linear combination of the vectors in [itex]S_{2}[/itex]. Then can you conclude that the two sets span the same space?

    If not, what if you further knew that every vector in [itex]S_{2}[/itex] is expressible as a linear combination of the vectors in [itex]S_{1}[/itex]?

    I merely need an answer. I will work out the details (proof) for myself. Thanks!!

    BiP
     
  2. jcsd
  3. May 4, 2013 #2

    jedishrfu

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    Isn't this like if every element of set A is in set B then there may still be elements in B that aren't in A.

    Geometrically if S1 is all vectors in a plane whereas S2 is all vectors in a 3D space then its clear that S2 has some vectors that aren't in S1.
     
  4. May 4, 2013 #3

    micromass

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    Yes, if you know both those things, then you can make the conclusion. If you only know one of these things, then jedishrfu already pointed out that the conclusion doesn't follow.
     
  5. May 4, 2013 #4

    WannabeNewton

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    No to the first and yes to the second. For the first one you can come up with tons of counter-examples. The second one you can prove straightforwardly from the definition of span.
     
  6. May 4, 2013 #5

    HallsofIvy

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    If every vector in S2 can be written as a linear combination of vectors in S1, then S2 spans some subspace of the span of S1.
     
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