$\langle Av,Aw\rangle=\langle V,w\rangle$,$\det(A)=1$ Proof Verification

Hi i don't know how to show that $\langle 2,x \rangle$ is not principal and the definition of a principal ideal is unclear to me. Feb 5, 2021i have what i think is a very simple question, basically, what does the notation "$\langle \rangle$" stand for? I was wondering what $|\langle a,b \rangle|$ mean, where both are vectors, if i am correct.

Jun 24, 2023what is the expectation value of $|\langle \psi|u|\psi \rangle|$ over haar random states $|\psi\rangle$? Apr 6, 2016i know that $\langle \phi | \psi \rangle$ is the probability of going from the $\psi$-state to the $\phi$-state, and that $\langle \phi | h | \phi \rangle$ is the expectation value of the energy fo.