Properties of length preserving linear transformations Let $T:\mathbb{R}^n\rightarrow\mathbb{R}^n$ be a linear transformation that preserves length. That is $||T(x)||=||x||$ for all $x\in\mathbb{R}^n$.Then: 


*

*If $\langle x,y\rangle=0$ then $\langle T(x),T(y)\rangle=0$.

*Show that columns of the matrix of $T$ in the standard basis of $\mathbb{R}^n$ are mutually orthogonal. 


For the answer
I have attempted using the basic definition of inner product with the norm.
That is $||v||^2=<v,v>$. Also with the Cauchy Schwarz inequality. But couldn't make any progress. 

The first claim follows readily from the polarization identity: $\langle x,y\rangle=\frac{1}{4}(\|x+y\|^2-\|x-y\|)$.
For the second claim, observe that the i-$th$ column of the matrix for $T$ is given by the coefficients on the right side of the equation $Te_i=\sum_ka_{ki}e_k.$ The result now follows by the first claim since 
$i\neq j\Rightarrow \langle e_i,e_j\rangle=0\Rightarrow \langle Te_i,Te_j\rangle=$
$\langle\sum_ka_{ki}e_k,\sum_ka_{kj}e_k \rangle =\sum_ka_{ki}a_{kj}=0,$ 
as desired. 
