The proof of the Lovász Sandwhich theorem, which says that the chromatic number of the complement of a graph $\chi(\bar{G})$ bounds the the Lovász number $\theta(G)$ is given here https://www2.math.ethz.ch/t3/fileadmin/math/ndb/00238/07403/Lovasz_eth_orth1.pdf
The definition given across many sources (also in Lovász's original paper) defines this Lovász number for some orthonormal representation $\left\{u_i:1\leq i \leq n \right\}$ as $$\theta(G)= \min_{c}\max_{1\leq i \leq n}\frac{1}{(c^{T}u_i)^2},$$ with $c$ ranging across all unit vectors.
In particular, in the proof the result of the sandwhich theorem follows from taking the handle vector $c$ to be the $1/\sqrt{k}(e_1+\dots+e_k)$ and associating the representation of $e_j$ to vertex $i$ when it is coloured with colour $j$.
However, in my spectral graph theory course, we are given that this number is defined only for the handle $e_1$, i.e. we define the Lovász number to be the infimum of all orthogonal representations of $\max_{i\in V} \frac{1}{(e_1^{T}u_i)^2}$, with $u_i$ the orthogonal representaion of vertex $i\in V$ and the lecturer told us that this trivially follows since a change of basis is sufficient to use either definition, but I don't see this and thus I don't see how the proof of this inequality follows from my given definition.
Suppose that you have an orthogonal representation $\{u_1,\ldots,u_n\}$ and some unit vector $c$. You can find some orthogonal transformation $A$ that sends $c$ to $e_1$. (This is easy: take an orthornomal basis $B_1$ containing $c$ and another orthonormal basis $B_2$ containing $e_1$, and define $A$ by mapping the basis $B_1$ to $B_2$.)
Orthogonal transformations preserve inner products, so in particular $\{Au_1,\ldots,Au_n\}$ is also an orthogonal representation and $$\langle c, u_i \rangle = \langle e_1, Au_i \rangle,$$ for each $i \in \{1,\ldots,n\}$. From this it should be easy to convince yourself that fixing an orthogonal representation and taking infimum over all unit vectors gives the same number as fixing a unit vector and taking infimum over all orthogonal representations.