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How to determine the smallest subspace?

  1. Jan 23, 2016 #1
    Two examples are:
    1. Given two subspaces ##U## and ##W## in a vector space ##V##, the smallest subspace in ##V## containing those two subspaces mentioned in the beginning is the sum between them, ##U+W##.
    2. The smallest subspace containing vectors in the list ##(u_1,u_2,...,u_n)## is ##\textrm{span}(u_1,u_2,...,u_n)##.
    How can one know how small a subspace is? initially I thought it was determined from the number of elements in the subspace, but there infinite number of elements. Can also someone please give an example by giving two subspaces and show the ways to compare which one is smaller than which?
     
  2. jcsd
  3. Jan 23, 2016 #2

    Samy_A

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    For 1:
    ##U+W## is the smallest subspace containing ##U## and ##W## means that if ##Z## is as subspace of ##V## with ##U \subseteq Z, W \subseteq Z##, then ##U+W \subseteq Z##.
    Similarly for 2. Any subspace containing ##(u_1,u_2,...,u_n)## will also contain ##\textrm{span}(u_1,u_2,...,u_n)##.

    Given two subpaces ##U, W##, you show that ##U## is smaller than ##W## by showing ##U \subset W##.
    It is of course possible that for two subspaces, neither is smaller than the other.
    Example in ##\mathbb R³##:
    ##U=\{(x,0,0)|x \in \mathbb R\}##,##W=\{(0,y,0)|y \in \mathbb R\}##. Neither subspace is a subset of the other, so there is no smallest of the two.
    ##U+W=\{(x,y,0)|x,y \in \mathbb R\}## is the smallest subspace of ##\mathbb R³## containing both ##U## and ##W##.
     
    Last edited: Jan 23, 2016
  4. Jan 23, 2016 #3
    It is the number of dimensions in the subspace. A line has one, a plane two, a solid three, etc.
     
  5. Jan 23, 2016 #4
    Thanks, that really makes sense.
    I get that ##U\subseteq U+W## and ##W\subseteq U+W##, therefore both ##U## and ##W## are smaller than ##U+W##. But I have a problem with convincing myself that the smallest subspace which contains ##U## and ##W## together is indeed ##U+W##, in other words, how can we be sure that there are no subspaces in ##U+W## which can contain both ##U## and ##W##. I am thinking that it may be because of the requirement of the closure under addition which must be true for a subset to be called subspace, but it's still hazy in my mind and I cannot sort what I am thinking into an ordered logical reasoning.
    Do you mean, the smallest subspace which contain ##U## and ##W## must have the same dimension as a subspace formed by adding the two subspaces, i.e. ##U+W##?
     
  6. Jan 23, 2016 #5

    Samy_A

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    (bolding mine)
    What I bolded in your post is indeed the key.
    A subspace that contains both ##U## and ##W## must contain all the sums of elements of ##U## and ##W## (as it must be closed under addition). In other words, it must contain ##U+W##. As ##U+W## is a subspace, we conclude it is the smallest subspace containing ##U## and ##W##.
     
  7. Jan 23, 2016 #6
    Alright thanks, at least now I am convinced that I have been going in the right direction in this matter.
     
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