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About vector space and subspaces

  1. May 17, 2012 #1
    Hi:
    I am a newbie to linear algebra; I have a problem about
    vector space and subspaces. How to distinguish these two
    subject. what I know from books is subspace is going through
    zero, but I still can not figure out what is the difference between
    vector space and subspaces, thanks.
     
  2. jcsd
  3. May 17, 2012 #2


    Simple: a vector subspace is a vector space in its own, but the particle "sub" indicates that it is a subset of a vector space that

    contains it, and they both are vectors spaces wrt the very same operations and, of course, over the same field.

    Thus, for example, the set [itex]\,\,\{t(1,0) \,;\,t\in\mathbb{R}\}\,[/itex] is a v. subspace of [itex]\,\mathbb{R}^2\,[/itex], but [itex]\,\mathbb{R}^2\,[/itex] is

    not a subspace of [itex]\,\mathbb{R}^3[/itex] is the former as not a subset of the latter.

    DonAntonio
     
  4. May 17, 2012 #3
    DonAntonio is correct, but I think his example is confusing.

    [itex]\mathbb{R}^2[/itex] is not a subspace of [itex]\mathbb{R}^3[/itex] because the former are vectors with two entries, and the latter are vectors with three entries. They are just different animals altogether, and so cannot be subsets of one or another.

    However, in practice we tend to think of [itex]\mathbb{R}^2[/itex] as a subset of [itex]\mathbb{R}^3[/itex] with the tacit understanding that every vector in [itex]\mathbb{R}^2[/itex] gets a zero appended to it.

    So [itex][4,2]\in\mathbb{R^2}[/itex] becomes [itex][4,2,0]\in\mathbb{R^3}[/itex]

    With this understanding, [itex]\mathbb{R}^2[/itex] is a subspace of [itex]\mathbb{R}^3[/itex]

    I add this not to nit pick or be confusing. I add this because this is how most people see [itex]\mathbb{R}^2[/itex] and it's important to make explicit why it is wrong to do so (like in DonAntonio's post) and what we implicitly do in our intuition.
     
    Last edited: May 17, 2012
  5. May 17, 2012 #4
    Thanks a lot
     
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