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Math Help - vector subspace

  1. #1
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    vector subspace

    If V={(x_1, x_2) ∈ R2|x_1≤0, x_2≤0} Is V a linear subspace of R2?


    Sol: I wrote, it's not a subspace since the vector -v= [-1,-1] belongs to the set but v does not belong to it. Am I right? Help!
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by Taurus3 View Post
    Am I right?

    Yes, you are.


    Fernando Revilla
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  3. #3
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    Just to elaborate a bit:

    V is not an \mathbb{R}-subspace of \mathbb{R}^2 because it is not closed under scalar multiplication by elements from the field \mathbb{R} (which is basically what your example is saying).

    That is, dealing with vector spaces over some field F, one of the conditions for V\subseteq W to actually be a subspace of W is that, for any \alpha \in \mathbb{R},v\in V, we must have \alpha v\in V.

    This is not the case in your example. As you pointed out, -1\in \mathbb{R} and <-1,-1>\in V, but -1v=<1,1> \notin V.
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