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Math Help - Proving a set is a subspace

  1. #1
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    Proving a set is a subspace

    Hey everybody,

    My tutor has found an old exam online and is having me complete it as a preparatory exercise before my final exam this week. I'm having trouble even wrapping my head around this one, so I'm hoping someone here can help me through it!

    (note: the ^ symbol is supposed to be a perpendicular symbol)

    Let W be a subspace of the inner product space V. Define
    W^ = {v in V: <v,w> = 0 for all w in W}
    (a) Prove that W^ is a subspace of V .

    If you'd like the test itself, its found at this address, problem number 12.a:

    http://www.math.niu.edu/courses/math...Fa04-final.pdf

    Thanks so much for any help!
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  2. #2
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    Quote Originally Posted by needsmathhelp101 View Post
    Hey everybody,

    My tutor has found an old exam online and is having me complete it as a preparatory exercise before my final exam this week. I'm having trouble even wrapping my head around this one, so I'm hoping someone here can help me through it!

    (note: the ^ symbol is supposed to be a perpendicular symbol)

    Let W be a subspace of the inner product space V. Define
    W^ = {v in V: <v,w> = 0 for all w in W}
    (a) Prove that W^ is a subspace of V .

    If you'd like the test itself, its found at this address, problem number 12.a:

    http://www.math.niu.edu/courses/math...Fa04-final.pdf

    Thanks so much for any help!
    The only way I know to prove something is a "subspace" is to use the definition of "subspace"! And that is "U is a subspace of V if and only if
    1) U is a subset of V
    2) U is closed under addition (the sum of two vectors in U is in U).
    3) U is closed under scalar multiplication (a number times a vector in U is in U).

    W^ is clearly a subset of V because the definition of W^ begins "{v in V...".

    Suppose u and v are both in W^. Then <u, w>= 0 and <v, w>= 0 for all w i W. What is <u+ v, w>?

    Suppose a is a scalar (number) and v is in W^. Then <v, w>= 0 for all w in W. What is <au, w>?
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