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Homework Help: Cauchy Schwarz Proof Question

  1. Mar 12, 2010 #1
    1. The problem statement, all variables and given/known data
    Prove that |x'y| <= ||x|| ||y|| for vectors x, y

    2. Relevant equations
    ||x|| is the norm of x
    x' is the transpose of x

    3. The attempt at a solution
    ||(x/||x||)-(y/||y||)|| = [ (x'/||x|| - y'/||y||)(x/||x|| - y/||y||) ]^1/2 = [-1/(||x||*||y||) (x'y + y'x) +2]^1/2
    we know that this norm is positive or zero, so:
    [-1/(||x||*||y||) (x'y + y'x) +2]^1/2 > 0
    -> x'y + y'x < 2||x||||y||
    x'y = y'x because they are scalars and the transpose of a scalar doesn't change its value
    -> 2x'y < 2||x||||y||
    -> x'y < ||x||||y||

    My result doesn't have absolute values around x'y; what am I doing wrong?
  2. jcsd
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