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U+v in subspace W, is u or v in subspace

  1. Jul 23, 2012 #1
    1. The problem statement, all variables and given/known data

    My question is if u+v is in the subspace can you say that u or v is in the subspace? If not would there be a counterexample?


    2. Relevant equations

    closed under addition/scalar multiplication

    3. The attempt at a solution

    I know that if u or v were in the subspace they would be closed under addition or multiplication. I don't know if you can say the same for (u+v) and apply it just to u or v.

    Thank you for any help.
     
    Last edited: Jul 23, 2012
  2. jcsd
  3. Jul 23, 2012 #2

    Mark44

    Staff: Mentor

    {0} is a subspace of R, a one-dimensional vector space. Are there vectors in R, that add to 0, that aren't in the subspace?

    BTW, we don't talk about vectors being closed under addition or scalar multiplication - we talk about the space they belong to as being closed under addition or scalar multiplication.
     
  4. Jul 23, 2012 #3
    Just making sure I have this correctly, that would mean that a or b is not in the vector space, just a+b. Thank you for the quick response.
     
  5. Jul 23, 2012 #4

    Mark44

    Staff: Mentor

    Don't think of a + b as being two things: it's a single thing. a and b are two vectors that happen to add up to whatever value a + b represents.
     
  6. Jul 24, 2012 #5

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    No, its as 0 dimension vector space. But your point is correct.

     
  7. Jul 24, 2012 #6

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    For example, the subset of R2, {(x, y)|y= x} is a subspace. The vectors (1, 0) and (1, 2) are not in that subspace but their sum, (2, 2), is.
     
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