$F:R^n$ to $R^n$ is a norm-preserving linear operator. Prove that $F(x)F(y)=xy$ for all $x,y$ in $R$

Asked Sep 08 '22 at 04:53

Active Sep 08 '22 at 04:53

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$F:R^n$ to $R^n$ is a norm-preserving linear operator. Prove that $F(x)*F(y)=x*y$ for all $x,y$ in $R$

Attempt: $||F(x+y)||^2=||x+y||^2=||F(x)||^2+||F(y)||^2+2x*y$

Then I'm stuck and don't know where to go.

asked Sep 08 '22 at 04:53

Skill HHY

1

You're close. Instead of swapping $|F(x)|$ and $|x|$ in one equation, write down two full equations of this form: One using only $F(x)$, $F(y)$ and $F(x+y)$ and the other using $x,y$ and $x+y$ – SBK Sep 08 '22 at 04:56
@T_M thanks! got it – Skill HHY Sep 08 '22 at 05:09

$F:R^n$ to $R^n$ is a norm-preserving linear operator. Prove that $F(x)*F(y)=x*y$ for all $x,y$ in $R$