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Norms and dot producks

  • #1

Homework Statement



(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.


Homework Equations



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


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
 

Answers and Replies

  • #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).
 
  • #3
jbunniii
Science Advisor
Homework Helper
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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:

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