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Math Help - Prove that P is a Subspace

  1. #1
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    Prove that P is a Subspace

    Hi all,

    Got asked the following question in lecture:

    Let \bf v \in R^n. Prove that P = \{\bf u \in  R^n: \bf v * \bf u = 0\} is a subspace of R^n.

    I understand that I must show that:
    1. P is not empty.
    2. P is closed under addition
    3. P is closed under scalar multiplication

    Just not sure, how I would show the 2nd and 3rd property. Thanks
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: Prove that P is a Subspace

    Quote Originally Posted by Oiler View Post
    Just not sure, how I would show the 2nd and 3rd property. Thanks
    I suppose * represents an inner product.

    If u_1,u_2\in P then, v*(u_1+u_2)=v*u_1+v*u_2=0+0=0 . Hence, u_1+u_2\in P . Try the third one.
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  3. #3
    MHF Contributor Siron's Avatar
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    Re: Prove that P is a Subspace

    U could also prove condition (2) and (3) together:
    Let u_1,u_2\in P and two scalars k,l\in \matbb{R} then, v* (ku_1+lu_2)=v* ku_1+v * lu_2=0 and so ku_1+lu_2 \in P
    Last edited by Siron; July 27th 2011 at 07:37 AM.
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