Suppose that $T:\mathbb{R}^n\rightarrow \mathbb{R}^n$ and suppose that $\{v_1,v_2,\cdots,v_n\}$ and $\{Tv_1,Tv_2,\cdots,Tv_n\}$ are orthonormal basis of $\mathbb{R}^n$. Prove that $T$ is a linear transformation.
Attempt:
I know that to show that $T$ is linear I must be able to show that given $u,v\in V$ I will have $T(u+v)=T(u)+T(v)$. I must also show that $T(cu)=cT(u)$.
I also know that I can write $u=\sum_i (v_i,v)v_i$ or $u=\sum_i(Tv_i,v)Tv_i$.
Now I'm not sure how to proceed. Could somebody give me an idea? Thank you!
From what you wrote, it does not follow that $T$ is a linear transformation. For example, if I have an orthonormal basis $\{v_1,v_2\dots,v_n\}$ for $\mathbb R^n$, then I can define $T$ as $$T(w) = \begin{cases}v_i&\text{ if } w=v_i\\0&\text{ otherwise}\end{cases}.$$
This $T$ fits all the demands you wrote about $T$, but it is not linear.