Example of inner product as matrix multiplication

Question

Can anyone please give an example for the following:

If I take $B=\{[1\ 0],[0\ 1]\} \implies M=I$. If $x=[1 \ 2], y=[3 \ 4]$, then $\langle x,y\rangle =11$. $[y]^H=[3 \ 4]^T$. But $[3 \ 4]^T M=[3 \ 4]^TI=[3 \ 4]^T \ne \langle x,y\rangle$? What am I doing wrong here?

Answer 1

You've calculated $y^H M$ instead of $y^H Mx$.

Answer 2

According to the argument you shared $M = [\mu_{lk}]$ where $\mu_{lk}= \langle v_k , v_l\rangle$. Since you chose your $B$ to be as such, then \begin{equation} v_1 = \begin{bmatrix} 1 \\ 0 \end{bmatrix} \quad \text{ and } \quad v_2 = \begin{bmatrix} 0 \\ 1 \end{bmatrix} \end{equation} So \begin{equation} M = \begin{bmatrix} \langle v_1,v_1\rangle & \langle v_1,v_2\rangle\\ \langle v_2,v_1\rangle & \langle v_2,v_2\rangle \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \end{equation}