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## Homework Statement

Let T be a linear operator on a finite dimensional vector space V. Suppose ||T(x)|| = ||x|| for all x in V, prove that T is one to one.

## Homework Equations

||T(x)||^2 = <T(x),T(x)>

||x||^2 = <x,x>

## The Attempt at a Solution

Suppose T(x) = T(y) x, y in V

Then ||T(x)|| = ||T(y)||

and so <T(x),T(x)> = <T(y),T(y)>

By assumption, we have <x,x> = <y,y>

But then i can't proceed on to show that x=y