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Math Help - Checking Vector Space Questions

  1. #1
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    Checking Vector Space Questions

    Okay, so I'm having some trouble with this topic. If a question asks you to show that a set is a subspace (or not a subspace), do you show the following:

    \bullet The set S is not empty
    \bullet The set is closed under vector addition, i.e. \mathbf{v,u}\in S \Longrightarrow \mathbf{v}+\mathbf{u}\in S
    \bullet The set is closed under scalar multiplication, i.e. \mathbf{u}\in S \ \mbox{and} \ \lambda \in \mathbb{F}\Longrightarrow \lambda\mathbf{u}\in S

    1) Show that the set

    S=\{\mathbf{x}\in\mathbb{R}^3:2x_1+3x_2-4x_3=6\}

    is not a subspace of \mathbb{R}^3.

    Would I do the following:

    2(\mathbf{0})+3(\mathbf{0})-4(\mathbf{0})=6

    Therefore, it does not contain the \mathbf{0} vector, hence not a subspace in \mathbb{R}^3
    Last edited by acevipa; September 8th 2010 at 10:26 PM.
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  2. #2
    A Plied Mathematician
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    Right, so if the zero vector is not included, you couldn't possible have closure under either scalar multiplication (think multiplication by zero) or vector addition (a vector plus its additive inverse should give you the zero vector). Generally yes, for a subspace, you get to inherit everything except the three properties you listed. So those are the ones you have to check.
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