(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.
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