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Games are having a weird issue lately. I’m not sure if it’s the. SKM Power Tools 6.5.1. (Full modules, 5000 buses with crack).Q:
Show $||x||=||y||$ implies $\langle x,x\rangle=\langle y,y\rangle$
Let $V$ be a complex linear space and $x,y\in V\setminus\{0\}$. Show that $||x||=||y||$ implies $\langle x,x\rangle=\langle y,y\rangle$.
The only thing that comes to my mind is to define a map $T:V\to\mathbb{C}$ by $T(x)=\langle x,x\rangle$ and then try to show that $T$ is injective. But I wasn’t able to find a way to prove this. Can somebody help me with this?
Thank you!
A:
Let $n=\dim(V)$. Consider the map
$$f:V\times V\to \mathbb{C}\\(v,u)\mapsto \langle v,u\rangle=\sum_{i=1}^nv_iu_i$$
Then $f$ is a Hermitian form. Then its condition:
$$f(x,x)=f(y,y)=\langle x,y\rangle=\langle y,x\rangle=f(x,y)^\ast$$
Means that the map $x\mapsto f(x,x)$ is an anisotropic linear form. But $x\mapsto f(x,x)$ is always anisotropic linear, because $(v_1+v_2,v_1+v_2)\geqslant 0$ and $(v_1+iv_
https://colab.research.google.com/drive/1rLSqZrReuPEaiptPWM-zVWTB2VBA5lCc
https://ello.co/tilpaolenpi/post/v1or0zlazk-avmgyiu3eng
https://colab.research.google.com/drive/1e6AWkieE51FcgNssv2mlJ9fNR1GdBXd2
https://ello.co/ancrocqcu_fu/post/e4j3pjo9nlrwa9xcs-hn9a
https://ello.co/diatiotio-ri/post/lt0jm4zqwy171shwjaiovq
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https://ello.co/elmutleacal/post/lsad_tmmdwasqeefe303gq
https://colab.research.google.com/drive/1t1zI9pcx8JuAtVZRfDBqpTLUlEt6Syn3
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