Given x,y in RR^n I am trying to prove this: ⟨x+y,x−y⟩=∥x∥_2^2−∥y∥_2^2

Lorelei Patterson 2022-07-16 Answered
Show that x + y , x y = x 2 2 y 2 2 ?
given x , y R n I am trying to prove this:
x y , x y = x , x 2 x , y + y , y = x 2 2 2 x , y + y 2 2 .
What I tried is using the fact that:
x y , x y = x , x 2 x , y + y , y = x 2 2 2 x , y + y 2 2 .
Then, since y=−(−y), I have
x + y , x y = x ( ) y , x y
But I don't know how I proceed from this... Maybe my approach is wrong?
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Answers (1)

escobamesmo
Answered 2022-07-17 Author has 18 answers
The simplest way to proceed is by using the bilinearity of the inner product, i.e., just expanding ⟨x+y,x−y⟩. Doing this:
x + y , x y = x , x y + y , x y = x , x x , y + y , x y , y = x 2 y 2
since the two middle terms cancel out (they are equal, inner product is commutative).

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