Let $\mathcal{B}$,$\mathcal{C}$ be the two orthonormal basis of inner product space $\mathbb{R}^n$.
Let $I:\mathbb{R}^n \to \mathbb{R}^n$ be the identity map and let $[I]_{\mathcal{B}}^{\mathcal{C}}\in M_{n \times n}(\mathbb{R})$ be the matrix corresponding to the linear map $I$ with respect to the bases $\mathcal{B}$ and $\mathcal{C}$ of $\mathbb{R}^n$.
Then is $[I]_{\mathcal{B}}^{\mathcal{C}}$ an orthogonal matrix?
I think it should be but I don't know how to prove it.
Would you shed me a light?
Consider the elements of $\mathbb{R}^n$ as column vectors and write $\mathcal{B} = (v_1, \dots, v_n), \mathcal{C} = (w_1, \dots, w_n)$. If you construct a matrix $B$ whose columns are the vectors $v_i$ then $B$ will be an orthogonal matrix (because the columns form an orthonormal basis). Similarly, you can construct the matrix $C$ whose columns are the vectors $w_i$ and it will be an orthogonal matrix. Then $[I]_{\mathcal{B}}^{\mathcal{C}} = BC^{-1}$ is orthogonal as a product of orthogonal matrices.