How can you show that an orthonormal basis can be expressed as the sum of a different orthonormal basis and the standard basis?

Question

Hi!

I understood from this question (Orthonormal basis question) that we have to show that an orthonormal basis can be expressed as the sum of a different orthonormal basis and the standard basis, in two different steps. I have tried to write it out, but I get stuck every time. Does anyone have an idea how to do this? Any help would be really appreciated, thank you in advance!

Answer 1

It's not the 'sum of a different orthonormal basis and the standard basis', but rather 'linear combination of another orthonormal basis'.

Well, it's not said, but the notation suggests, and if it feels better, we can calmly assume that $e_1,e_2,\dots,e_n$ is the standard basis of $\Bbb R^n$.
Nevertheless, it's enough to assume that it's an orthonormal basis.

For 1. simply use that $e_1,e_2,\dots,e_n$ is a basis, and hence any vector - and thus in particular the vectors $f_k$ - can be expressed by a unique linear combination of them.

For 2., note that the $i$th column ${\bf q}_i=(q_{i1},\dots,q_{in})^T$ of $Q$ is the coordinate vector of $f_i$ coordinatized in $(e_1,\dots,e_n)$, and prove that $\langle {\bf q}_i,\,{\bf q}_j\rangle\,=\,\langle f_i,f_j\rangle$.