Few basic things unclear to me about inner product spaces and orthonormal basis

Question

Few things unclear to me about inner product spaces:

assume V is an inner product space with B orthonormal basis. Why is it true that:

1. $$\langle x,y\rangle = \langle[x]_{B} , [y]_B \rangle{st}$$ when the first is some inner product and the second has st on it to mark that it is the standard inner product

2. $$[Ci]^{*}_{B} [Cj]_{B} = \langle[Ci]_{B}, [Cj]_{B} \rangle{st}$$ when Ci and Cj are columns in a matrix

thank you very much!

Answer 1

Concering your first question:

If, say $B = \{ b_1, \dots, b_n \}$, $x = \sum_{i=1}^n a_i e_i$ and $y = \sum_{i=1}^n b_i e_i$ and if you mean by $[x]_B$ the tuple $(a_1, \dots, a_n)$ and by $[y]_B$ the tuple $(b_1, \dots, b_n)$, then $\langle x, y \rangle = \langle \sum_{i=1}^n a_i e_i, \sum_{j=1}^n b_j e_j \rangle = \sum_{i=1}^n \sum_{j=1}^n a_i b_j \langle e_i, e_j \rangle = \sum_{i=1}^n \sum_{j=1}^n a_i b_j \delta_{i,j} = \sum_{i=1}^n a_i b_i = \langle [x]_B, [y]_B \rangle_{st}$.

As for the second, if $C = (c_{ij})_{i,j}$, then $[Ci] = (c_{1i}, \dots, c_{ni})$ and $[Cj] = (c_{1j}, \dots, c_{nj})$.

Then you get $[Ci]^*[Cj] = \sum_{k=1}^n c_{ki} c_{kj} = \langle[Ci],[Cj]\rangle_{st}$

Remark: I suppressed the subindex $B$ here, since I do not see any reason for it.