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

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1 Answer 1

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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.

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

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