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Norms and dot producks

  1. May 18, 2009 #1
    1. The problem statement, all variables and given/known data

    (a) Using the dot product, show that for x, y ∈ Rn, the formula
    2||x||^2 + 2||y||^2 = ||x + y||^ 2 + ||x − y||^2 holds.

    (b) The norm on Rn can be defined in terms of the dot product by
    the formula ||x|| = √(x • x). Show that the reverse is true. That is,
    find a formula for x • y involving the norms of vectors (||x||, ||y||,
    ||x+y||, and ||x−y|| for example), and without using coordinates.


    2. Relevant equations

    http://upload.wikimedia.org/math/e/9/0/e901236dd90219902035e3479b31a2ee.png


    3. The attempt at a solution

    for part a, i started with the right side
    (i use the word sum to represent the summation from 1 to n of each variable
    ||x+y||^2 + ||x-y||^2 = (x+y)•(x+y) + (x-y)•(x-y) = sum[(x+y)(x+y)] + sum[(x-y)(x-y)] = sum x^2 +sum (2xy) + sum y^2 + sum x^2 - sum (2xy) + sum y^2 = 2sumx^2 + 2sumy^2 = 2||x||^2 + 2||y||^2 = left side

    i'm pretty sure im doing this part right but it never hurts to double check

    now part b i just have no idea where to start with
    im supposed to express the dot product x•y in terms of norms .... but x•y= sum from i=1 to n of (xi*yi) = x1y1 + x2y2+......+xnyn
    I can't think of anything to factor or do to that last expression in order to get it looking like something I can express in terms of norms
    i know that x•y = ||x|| ||y|| cos(theta) where theta is the angle between x and y, but i think i need a way to express it using only norms
     
  2. jcsd
  3. May 18, 2009 #2

    Cyosis

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    A is correct but you don't need to use sums. The inner product is distributive so you can say [itex](x+y,x+y)=(x,x)+(y,y)+2(x,y)[/itex]. Saves some time.

    I find b a bit vague but you could calculate (x+y,x+y) and (x-y,x-y) and solve for (x,y).
     
  4. May 18, 2009 #3

    jbunniii

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    (b) is indeed a bit vague. There are two possible interpretations that come to mind:

    (1) Assuming that the norm is induced by a dot product, i.e.,

    [tex]||v|| = <v,v>^{1/2}[/tex]

    then find an expression for [tex]<x,y>[/tex] in terms of the norm. This isn't too hard but involves a bit of trial and error to find something that works.

    (2) Assuming only that you are given a norm, prove that there exists a dot product that induces it, and find a formula for it.

    Of note is that (2) is NOT TRUE in general. In fact, it happens to be possible for a given norm IF AND ONLY IF that norm satisfies the parallelogram law, i.e., part (a) of the OP's question.

    Somewhat incongruously, (2) is given as an exercise in Axler's "Linear Algebra Done Right." The only proof I've been able to find is at least an order of magnitude harder than the level of the typical exercise in that book. I posted on that very subject this past weekend:

    https://www.physicsforums.com/showthread.php?t=314503

    Perhaps someone knows of a more elementary solution.
     
    Last edited: May 18, 2009
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