Let x,y be vectors in$\mathbb R$. Show that for any real number λ.

Question

$λ∥x∥^2 +(1−λ)∥y∥^2 =∥λx+(1−λ)y∥^2 +λ(1−λ)∥x−y∥^2$

I don't understand what to do with this?

I have the properties:

$∥x+y∥^2=∥x∥^2+∥y∥^2+2<x,y>$

$<x,y>=<y,x>$

to work with but (1−λ) prevents me from doing anything here because it creates an additional y that I can't remove.

Answer 1

Start with the right side. $\|\lambda x+(1-\lambda)y\|^{2}+\lambda (1-\lambda)\|x-y\|^{2}=[\lambda^{2}\|x\|^{2}+(1-\lambda)^{2} \|y\|^{2}+2\lambda (1-\lambda)\langle x , y \rangle ] +\lambda (1-\lambda) [\|x\|^{2}+\|y\|^{2}-2\langle x , y \rangle ]$.

Now just simplify this.