Ex5 1

$\def\abs#1{|#1}\def\i{\mathbf {i}}\def\ket#1{|{#1}\rangle}\def\bra#1{\langle{#1}|}\def\braket#1#2{\langle{#1}|{#2}\rangle}\mathbf{Exercise\ 5.1}$

Show that any linear transformation $U$ that takes unit vectors to unit vectors preserves orthogonality: if subspaces $S_1$ and $S_2$ are orthogonal, then so are $U S_1$ and $U S_2$.

page revision: 0, last edited: 18 Nov 2012 18:35

If $U$ takes unit vectors to unit vectors it preserves the inner product because $<x|x> = 1=<x|U^\dagger U|x>$

So then if $|x>$ and $|y>$ are orthogonal then $<x|y>$ = 0 and $<x|U^\dagger U|y> = 0$ as well.

Not sure if this is sufficient but best I came up with so far.

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