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Math Help - Subspace of R^3

  1. #1
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    Subspace of R^3

    Hello,

    I was wondering if \mathbb{R}^2 is a subspace of \mathbb{R}^3? Some classmates of me tell me it is, some say it isn't.

    I'm under the impression that it is in fact a subspace, anyone care to elaborate?

    Thanks in advance
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  2. #2
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    Sure it is. \mathbb{R}^{2} is a vector space in its own right, and it is a subset of \mathbb{R}^{3}. Ergo, it's a subspace. The only confusion might be in a mix of notations. But you can just say, for example, that the third component is always zero (sort of an embedding scheme, I guess).

    Does that help?
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  3. #3
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by chris2547 View Post
    I was wondering if \mathbb{R}^2 is a subspace of \mathbb{R}^3? Some classmates of me tell me it is, some say it isn't.
    A necessary condition to be F subspace of E is that F\subset E. Taking into account that \mathbb{R}^2\not\subset \mathbb{R}^3 we can't say strictly that \mathbb{R}^2 is a subspace of \mathbb{R}^3.

    However (and following Ackbeet's outline) you can define:

    f:\mathbb{R}^2\rightarrow \mathbb{R}^3\;\quad f(x_1,x_2)=(x_1,x_2,0)

    This function is an monomorphism (with usual operations) , so we can identify algebraically \mathbb{R}^2 with \textrm{Im}f\subset \mathbb{R}^3 (which is a subspace of \mathbb{R}^3. In that sense we say " \mathbb{R}^2 is a subspace of \mathbb{R}^3 " .

    Fernando Revilla
    Last edited by FernandoRevilla; December 28th 2010 at 06:53 AM.
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  4. #4
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    Just to clarify what Ackbeet is saying:

    Technically it is not a subspace, but it is isomorphic to one in a very strong sense (by adding an extra component which is always zero).

    Most mathematicians will say it is a subspace by thinking in terms of this isomorphism (they are technicaly wrong by saying this, but this convention has been agreed upon in the mathematical community).
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