# Does $\langle Tv, w \rangle = \langle v, w \rangle$ imply that $T = I$?

by Jayanth Rao   Last Updated August 14, 2019 09:20 AM - source

Suppose $$V$$ is an inner product space and $$T \in \mathcal{L}(V)$$. Suppose $$\langle Tv, w \rangle = \langle v, w \rangle$$ for all $$v, w \in V$$. Does this imply that $$T = I$$?

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#### Answers 3

$$\langle Tv,w\rangle-\langle v,w\rangle=\langle Tv-v,w\rangle=0$$ for any $$v,w\in V$$. Hence $$Tv-v=0$$ for any $$v\in V$$ hence $$T=Id$$.

InsideOut
August 14, 2019 09:14 AM

Yes Just take $$w=Tv-v$$ to see that $$\|Tv-v\|^{2}=0$$ so $$Tv=v$$.

Kavi Rama Murthy
August 14, 2019 09:14 AM

Yes. $$\forall v,w\quad \langle Tv,w\rangle = \langle v,w\rangle \Leftrightarrow \\ \forall v,w\quad \langle (T-I)v,w\rangle = 0 \Leftrightarrow \\ \forall v\quad (T-I)v = 0 \Leftrightarrow \\ T-I =0$$

Adam Latosiński
August 14, 2019 09:15 AM