Let $v_1,...,v_n$ and $w_1,...w_n$ be two sets of linearly independent vectors in $\mathbb{R^n}$. Show that all their dot products are the same, so $v_j \dot\ v_i = w_i \dot\ w_j$ for all $i,j = 1,...,n$ iff there is an orthogonal matrix $Q$ such that $w_i = Qv_i$ for all $i=1,...,n$.

My attempt:

$\langle v+w,v+w\rangle = \langle v,v\rangle+\langle v,w\rangle+\langle w,v\rangle+\langle w,w\rangle$ and since $w_i = Qv_i$ we have that $\|v\|^2 = \|w\|^2$ ?


1 Answer 1


A matrix is orthogonal iff $$ \langle Qx,Qy \rangle = \langle x,y \rangle $$ for all $x$, $y\in \mathbb{R}^n$. Since both $\{x_1,\ldots,x_n\}$ and $\{y_n,\ldots,y_n\}$ are bases of $\mathbb{R}^n$, there exists an invertible matrix $Q$ mapping $x_j$ to $y_j$, for all $j$. Now you should be able to conclude.

  • $\begingroup$ Grazie tanto!!! $\endgroup$
    – diimension
    Oct 25, 2012 at 23:40

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.