Question

Prove that a linear transformation is norm preserving if and only if it is inner product preserving.

Answer 1

@MathSofiya

Answer 2

I was able to prove it one direction, starting with it being inner product preserving and showing that it was norm preserving. That was pretty easy. The other direction is confusing me, though.

Answer 3

Basically we want to show: \(|Tx|=|x|\implies \left<Tx,Ty\right>=\left<x,y\right>\).

Answer 4

Also, I want to show that all norm-preserving linear transformations are one-to-one.\[
|Tx|=|x|\implies(Tx=Ty\implies x=y)
\]

Answer 5

Here's another way to write what we want to show: \[
\sqrt{\sum Tx_i^2}=\sqrt{\sum x_i^2}\implies \sum Tx_iTy_i=\sum x_iy_i
\]All the sums are from i=1 to n assuming these are n-dimensional vectors.

Answer 6

Okay, here's an idea. Using the linearity of the transformation we know that T(x+y)=Tx+Ty, so that means that |Tx+Ty|=|x+y| and |Tx-Ty|=|x-y|, and we know that \[\left<x,y\right>=\frac14(|x+y|^2-|x-y|^2)
\]I think that does it, actually.