Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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 $\|v\|^2 = \|w\|^2$ ?

share|improve this question
add comment

1 Answer 1

up vote 2 down vote accepted

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.

share|improve this answer
    
Grazie tanto!!! –  diimension Oct 25 '12 at 23:40
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

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