Orthogonal Transformation Let $L$ be a transformation on a finite-dimensional real vector space $V$ with an inner product. Also $L$ sends each orthonormal basis in $V$ to another orthonormal basis. Prove that $L$ is an orthogonal transformation.
I have trouble proving the first step which is trying to prove that this transformation is linear. How should I continue? Any hints?
 A: Let $v=\sum v_i e_i$ (basically a basis expansion of v). Then: $Lv = L(\sum v_i e_i)$. Use the IP against some $e_j$:
$$\left\langle L(\sum v_i e_i), e_j\right\rangle \\ = \left\langle \sum v_i e_i, L^*(e_j)\right\rangle \text{adjoint always exists in finite dims} \\ = \sum v_i \left\langle  e_i, L^*(e_j)\right\rangle \text{linearity of IP} \\ = \sum v_i \left\langle   L(e_i), e_j\right\rangle \\ = \sum \left\langle  v_i L( e_i), e_j\right\rangle \\ = \left\langle  \sum v_i L( e_i), e_j\right\rangle $$
This is true for all $e_j$ so $L\left(\sum v_i e_i\right) = \sum v_i L( e_i)$
A: Let be $A = \{e_1,...,e_n\}$ ortonormal base of $V$. By hipotesis $B =\{Le_1,...Le_n\}$ it's an ortonormal base of $V$, therefore, if $a_{ij} \in R$ are such that 
$$Te_j = \sum_{i=1}^{n}a_{ij}e_i$$
we have from ortogonalyties of $A$ and $B$ that 
$$\delta_{jk} = \langle Te_j,Te_k \rangle = \sum_{i = 1}^n a_{ij}a_{ik} = \sum_{i = 1}^n a_{ki}^Ta_{ij} $$
and therefore, $(a_{ki}^T)\times (a_{jl}) = I$, where $(a_{ki}^T)$ is transpost matrix of $(a_{jl})$, $I$ identity matriz of $n\times n$ ordem. Because $(a_{jl})$ is matrix of $L$ on terms of $A$, we have $L$ is ortogonal.
