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 $T = I$?



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
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
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
Adam Latosiński
August 14, 2019 09:15 AM

Related Questions


linear equations elimination method help

Updated February 28, 2018 19:20 PM



Linear combinations of intervals

Updated June 01, 2015 11:08 AM

Compute the standard matrix of P to V.

Updated May 01, 2017 14:20 PM