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Span is a subspace proof

  1. Apr 6, 2009 #1
    I have a problem.
    Suppose that {u1,u1,...,um} are vectors in R^n. Prove, dircetly that span{u1,u2,...,um} is a subspace of R^n.
    How would I go by doing this?
  2. jcsd
  3. Apr 6, 2009 #2


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    Well, directly, as the question asks. Where are you stuck?
  4. Apr 6, 2009 #3
    Depends on your definition of span (my favourite being span {u1, ..., um} = the smallest subspace containing u1, ..., um, from which the result is trivial). ;)

    You probably define span {u1, ..., um} = {a1u1 + ... + amum | a1, ..., am in R}. Just apply your definition of (or test for) a subspace.
  5. Apr 6, 2009 #4
    I just need to know how to get started.
  6. Apr 6, 2009 #5
    Well, your span (probably meaning as adriank pointed out) is the the set of all linear combinations of those vectors. So that's clearly a subset of your vector space, right?

    So, what's the difference between a subspace of a vector space and just a plain old subset? What's the magic property that spaces have that sets don't? Then you just need to demonstrate that your subset has it.
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