Inner product space, orthonormal bases and change of basis.

Question

I define unitary as $B*B=I$ I know that part (i) requires me to show the matrix coefficients are that of the inner product for bases A and B, however I am unsure how to get to this.

Any help would be much appreciated in the lead up to my examination.

Answer 1

Expressing $w_i$ in terms of the basis $B$ we have \begin{equation} w_i=\sum_{j=1}^n \langle w_i,v_j \rangle v_j \end{equation} (do you know why this is true, or must I elaborate?),

and since $C_{AB}=[[w_1]_B,\ldots,[w_n]_B]$ you can see that the i'th column of $C_{AB}$ will then be \begin{equation} \begin{bmatrix} \langle w_i,v_1 \rangle \\ \vdots \\ \langle w_i,v_n \rangle \end{bmatrix}. \end{equation}

Then again for part (ii), if you want to do it in detail you can show \begin{equation} 1=\|w_i\|^2=\sum_{j=1}^n \langle w_i,v_j \rangle^2 \text{ and } 0=\langle w_i,w_k\rangle =\sum_{j=1}^n \langle w_i,v_j \rangle \langle w_k,v_j \rangle, \end{equation} just by using the properties of the orthonormal bases, from which it follows that $C_{AB}^*C_{AB}=I$.