Let's assume $v,w, x_i \in R^n$ are unknown. Can one compute dot product $\langle v,w\rangle$ if one has just the numbers: $\langle v,x_i\rangle$ and $\langle w,x_i\rangle$ for $n$ random vectors $x_i$.

If $x_i = e_i$ it is quite simple: $$ \langle v,w\rangle = \sum_i \langle v,e_i\rangle \langle w,e_i\rangle$$

But what if the $x_i$ is not an orthogonal basis ?
If not possible can we do something with stronger assumptions like all vectors being unitary ?

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    $\begingroup$ You must assume the vectors $x_i$ form a basis. Also, one can express the vector $v$ as a linear combination of the $x_i$'s and use linearity to obtain an expression. So, you probably mean to ask whether the product $\langle v,w\rangle$ can be expressed in terms of the numbers $\langle v,x_i\rangle$ and $\langle w,x_i\rangle$ only. $\endgroup$ Apr 26, 2016 at 8:30
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    $\begingroup$ I think you'll also need information on $<x_i,x_j>$. $\endgroup$ Apr 26, 2016 at 9:05
  • $\begingroup$ You might want to look at frames. That would be a case when you give more than $n$ $x_i$ vectors, but based on the $\langle v , x_i \rangle$ and $\langle w , x_i \rangle$. $\endgroup$
    – AHusain
    Apr 26, 2016 at 9:21


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