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Math Help - [SOLVED] Positive Definite Inner Product proofs

  1. #1
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    [SOLVED] Positive Definite Inner Product proofs

    Let V be a vector space and < , > be a positive definite inner product. Then show that:
    a) < 0, v > = 0 for any v in V
    b) Show that for a fixed u in V, < u, v > = 0 for any v in V, then u = 0.
    c) Show that < v, w > = 0 for any w in V implies that v = 0.

    I did part a) like this:
    < 0, v > = < 0 + 0, v > = < 0, v > + < 0, v>
    < 0, v > = < 0, v > + < 0, v >
    < 0, v > - < 0, v > = < 0, v >
    0 = < 0, v>

    Now I have been trying to prove parts b) and c) but I cannot think of a way to start. The statements both seem so obvious but I don't know where to begin with a proof.
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  2. #2
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    Quote Originally Posted by Zalren View Post
    Let V be a vector space and < , > be a positive definite inner product. Then show that:
    a) < 0, v > = 0 for any v in V
    b) Show that for a fixed u in V, < u, v > = 0 for any v in V, then u = 0.
    In particular, take v= u.

    c) Show that < v, w > = 0 for any w in V implies that v = 0.
    Same suggestion- take w= v.

    I did part a) like this:
    < 0, v > = < 0 + 0, v > = < 0, v > + < 0, v>
    < 0, v > = < 0, v > + < 0, v >
    < 0, v > - < 0, v > = < 0, v >
    0 = < 0, v>

    Now I have been trying to prove parts b) and c) but I cannot think of a way to start. The statements both seem so obvious but I don't know where to begin with a proof.
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  3. #3
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    Thanks! I feel like it was too easy once you helped me with the first step. I was just stuck for some reason.
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