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Help proving a subset is a subspace

  1. Apr 16, 2009 #1
    1. The problem statement, all variables and given/known data

    Prove that the set of all 3-vectors orthogonal to [1, -1, 4] forms a subspace of R^3.

    2. Relevant equations

    Orthogonal means dot product is 0.

    3. The attempt at a solution

    I know the vectors in this subspace are of the form
    [a,b,c] where a - b + 4c = 0.
    However I don't know how to use this to show there is closure under vector addition and scalar multiplication.
  2. jcsd
  3. Apr 16, 2009 #2


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    Homework Helper
    Gold Member

    Let v_1 and v_2 be two vectors orthogonal to the given vector (call it a), i.e.

    [tex] v_1 \cdot a = 0 [/tex]
    [tex] v_2 \cdot a = 0 [/tex]

    Now, using these, all you have to do is show that

    [tex] (v_1 + v_2) \cdot a = 0[/tex]

    HINT: Use distributivity of the dot product.
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