I'm struggling with finding info about how to calculate tensor products so I'm having problems with this exercise. (My teachers never told us anything about it and assume we know how it works.)
I searched on internet information but couldn't find anything that would help me out. It's a quantum optics exercise but has a mathematics question. The set:
$\left\{ \begin{pmatrix} 0 \\ 1 \\ \end{pmatrix} \otimes |n\rangle,\begin{pmatrix} 1 \\ 0 \\ \end{pmatrix} \otimes |n-1\rangle, n=1,2,...\right\}$ is a basis of the Hilbert space. Show that it is an orthonormal basis.$$ Nb: â^{\dagger}â|n\rangle=n|n\rangle; N=â^{\dagger}â + 1/2 \begin{pmatrix} 1 & 0 \\ 0 & -1\\ \end{pmatrix}; $$ I don't know where to start at all. I do know that orthonormal means orthogonal and that the norm is 1. I don't know how to prove the orthogonality beside calculating a scalar product that should be equals to zero.